A073405 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n >= 0, (generalized (2,2)-Fibonacci). Companion triangle is A073406.

1, 36, 12, 1536, 888, 120, 80448, 62592, 15168, 1152, 5068800, 4813056, 1600704, 222336, 10944, 375598080, 413351424, 169917696, 32811264, 2992896, 103680, 32103751680, 39661608960, 19066503168, 4592982528

Offset

0,2

Comments

The row polynomials are p(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of U0(n) := A002605(n), n >= 0, ((2,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073387(n+k,k) = 2*(p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*U0(n))/(k!*12^k)), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073406(k,m).

Links

Table of n, a(n) for n=0..24.

W. Lang, First 7 rows.

Formula

Recursion for row polynomials defined in the comments: p(k, n)= 2*(2*(n+2)*p(k-1, n+1)+2*(n+2*(k+1))*p(k-1, n)+(n+3)*q(k-1, n+1)); q(k, n)= 4*((n+1)*p(k-1, n+1)+(n+2*(k+1))*q(k-1, n)), k >= 1. [Corrected by Sean A. Irvine, Nov 25 2024]

Example

k=2: U2(n)=2*((36+12*n)*(n+1)*U0(n+1)+(36+12*n)*(n+2)*U0(n))/(2!*12^2), cf. A073389.
Triangle begins:
  1;
  36, 12;
  1536, 888, 120;
  ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Crossrefs

Cf. A002605, A073387, A073406, A073403.

Sequence in context: A109256 A277983 A066583 * A298572 A260383 A056770

Adjacent sequences: A073402 A073403 A073404 * A073406 A073407 A073408

Keyword

nonn,easy,tabl,changed

Author

Wolfdieter Lang, Aug 02 2002

Status

approved