A154229 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)^2*T(n-2, k-1), read by rows.

1, 1, 1, 1, 38, 1, 1, 139, 139, 1, 1, 365, 8828, 365, 1, 1, 807, 70492, 70492, 807, 1, 1, 1592, 357459, 7062136, 357459, 1592, 1, 1, 2889, 1404923, 98777227, 98777227, 1404923, 2889, 1, 1, 4915, 4631612, 824036625, 14498379854, 824036625, 4631612, 4915, 1

Offset

0,5

Comments

Row sums are: {1, 2, 40, 280, 9560, 142600, 7780240, 200370080, 16155726160, ...}.

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) = binomial(n+2, 2)^2 = A000537(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)*(n+2)/2)^2*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Example

Triangle begins as:
  1;
  1,    1;
  1,   38,       1;
  1,  139,     139,         1;
  1,  365,    8828,       365,           1;
  1,  807,   70492,     70492,         807,         1;
  1, 1592,  357459,   7062136,      357459,      1592,       1;
  1, 2889, 1404923,  98777227,    98777227,   1404923,    2889,    1;
  1, 4915, 4631612, 824036625, 14498379854, 824036625, 4631612, 4915, 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) + binomial(n+2, 2)^2*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] + Binomial[n+2, 2]^2*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
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 >;
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, A154230, A154231, A154233.

Cf. A000537.

Sequence in context: A023930 A022072 A351221 * A225433 A225398 A037936

Adjacent sequences: A154226 A154227 A154228 * A154230 A154231 A154232

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 05 2009

Extensions

Edited by G. C. Greubel, Mar 02 2021

Status

approved