A154231 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12)*T(n-2, k-1), read by rows.

1, 1, 1, 1, 278, 1, 1, 1579, 1579, 1, 1, 6005, 1233308, 6005, 1, 1, 18207, 20504692, 20504692, 18207, 1, 1, 47216, 194715939, 35816807848, 194715939, 47216, 1, 1, 108993, 1319518787, 1302709376779, 1302709376779, 1319518787, 108993, 1, 1, 229819, 7024500980, 24830582225241, 4330171226988158, 24830582225241, 7024500980, 229819, 1

Offset

0,5

Comments

Row sums are: {1, 2, 280, 3160, 1245320, 41045800, 36206334160, ...}.

The row sums of this class of sequences (see cross-references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = (n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12 = A000539(n+1). - G. C. Greubel, Mar 02 2021

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2 +6*n +3)/12)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Example

Triangle begins as:
  1;
  1,      1;
  1,    278,          1;
  1,   1579,       1579,             1;
  1,   6005,    1233308,          6005,             1;
  1,  18207,   20504692,      20504692,         18207,          1;
  1,  47216,  194715939,   35816807848,     194715939,      47216,      1;
  1, 108993, 1319518787, 1302709376779, 1302709376779, 1319518787, 108993, 1;

Maple

T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else T(n-1, k) + T(n-1, k-1) + ((n+1)^2*(n+2)^2*(2*n^2+6*n+3)/12)*T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + ((n+1)^2*(n+2)^2*(2*n^2+6*n+3)/12)*T[n-2, k-1] ];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)

Prog

(Sage)
def f(n): return binomial(n+2, 2)^2*(2*n^2+6*n+3)/3
def T(n, k):
    if (k==0 or k==n): return 1
    else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021
(Magma)
f:= func< n | Binomial(n+2, 2)^2*(2*n^2+6*n+3)/3 >;
function T(n, k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
  end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021

Crossrefs

Cf. A154227, A154228, A154229, A154230, A154233.

Cf. A000539 (powers of 5).

Sequence in context: A048525 A189609 A264371 * A252249 A257368 A056995

Adjacent sequences: A154228 A154229 A154230 * A154232 A154233 A154234

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Jan 05 2009

Extensions

Edited by G. C. Greubel, Mar 02 2021

Status

approved