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bibkey: kimberling2004a054096 authors: Clark Kimberling; Ralf Stephan year: 2004 title: “OEIS A054096, T(n,2), array T as in A054090” doi: null url: https://oeis.org/A054096 claim: “%N T(n,2), array T as in A054090. %C Conjecture: sequence is A006183 shifted right. - Ralf Stephan, Jan 15 2004. %N A054090: Triangular array generated by its row sums: T(n,0) = 1 for n >= 0, T(n,1) = r(n-1), T(n,k) = T(n,k-1) - (-1)^k * r(n-k) for k = 2, 3, …, n, n >= 2, r(h) = sum of the numbers in row h of T. %F A054090: T(n, k) = T(n, k-1) - (-1)^k * Sum_{j=0..n-k} T(n-k, j), with T(n, 0) = 1, and T(n, 1) = Sum_{j=0..n-1} T(n-1, j). %N A006183: a(n) = (n+1)*a(n-1) + (2-n)*a(n-2).” strata_touched:

  • D5/S1/Recurrence/StephanTriangleSecondColumn license: citation-only triage: anchor

OEIS A054096

The NAME of A054096 states:

T(n,2), array T as in A054090.

Ralf Stephan’s target COMMENT states:

Conjecture: sequence is A006183 shifted right. - Ralf Stephan, Jan 15 2004

The NAME of A054090 states:

Triangular array generated by its row sums: T(n,0) = 1 for n >= 0, T(n,1) = r(n-1), T(n,k) = T(n,k-1) - (-1)^k * r(n-k) for k = 2, 3, …, n, n >= 2, r(h) = sum of the numbers in row h of T.

Its FORMULA line states:

T(n, k) = T(n, k-1) - (-1)^k * Sum_{j=0..n-k} T(n-k, j), with T(n, 0) = 1, and T(n, 1) = Sum_{j=0..n-1} T(n-1, j).

The NAME of A006183 states:

a(n) = (n+1)*a(n-1) + (2-n)*a(n-2).

The AUTHOR line of A054096 is:

Clark Kimberling

Clark Kimberling authored the sequence, while Ralf Stephan authored the target conjectural comment. The AUTHOR line is undated; the bibliographic year 2004 records the date of Stephan’s conjecture.

Verified locator

  • URL: https://oeis.org/A054096
  • A054096 NAME (%N, verbatim): T(n,2), array T as in A054090.
  • A054096 COMMENT (%C, verbatim): Conjecture: sequence is A006183 shifted right. - Ralf Stephan, Jan 15 2004
  • A054090 NAME (%N, verbatim): Triangular array generated by its row sums: T(n,0) = 1 for n >= 0, T(n,1) = r(n-1), T(n,k) = T(n,k-1) - (-1)^k * r(n-k) for k = 2, 3, …, n, n >= 2, r(h) = sum of the numbers in row h of T.
  • A054090 FORMULA (%F, verbatim): T(n, k) = T(n, k-1) - (-1)^k * Sum_{j=0..n-k} T(n-k, j), with T(n, 0) = 1, and T(n, 1) = Sum_{j=0..n-1} T(n-1, j).
  • A006183 NAME (%N, verbatim): a(n) = (n+1)*a(n-1) + (2-n)*a(n-2).
  • A054096 AUTHOR (%A, verbatim): Clark Kimberling

A054096 has offset 2 and A006183 has offset 1, so “shifted right” means A054096(n) = A006183(n-1) for every n >= 2. The %S and %T data lines of the two entries were checked term for term and agree under this shift.