OFFSET
0,5
COMMENTS
Row sums are: {1, 2, 4, 28, 310, 3588, 45236, 642276, 10312214, 185760988, 3715773650, ...}.
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 2, 1;
1, 13, 13, 1;
1, 58, 192, 58, 1;
1, 209, 1584, 1584, 209, 1;
1, 682, 10335, 23200, 10335, 682, 1;
1, 2125, 60267, 258745, 258745, 60267, 2125, 1;
1, 6482, 330942, 2482938, 4671488, 2482938, 330942, 6482, 1;
MAPLE
b:= binomial; T:= 2 + sum((-1)^j*b(n+1, j)*(2*(k-j)+1)^n, j=0..k) - 2*b(n, k)*b(n+1, k)/(k+1); seq(seq(T(n, k), k = 0 .. n), n = 0 .. 10); # G. C. Greubel, Nov 23 2019
MATHEMATICA
(* First program *)
p[x_, n_]= (1 - x)^(n+1)*Sum[(2*k+1)^n*x^k, {k, 0, Infinity}]; (*A060187*)
f[n_, m_]:= CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m+1]];
T[n_, m_]:= 2 -(-f[n, m] +2*Binomial[n, m]*Binomial[n+1, m]/(m+1));
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
(* Second program *)
B:=Binomial; T[n_, k_]:= T[n, k]= 2 +Sum[(-1)^j*B[n+1, j]*(2*(k-j)+1)^n, {j, 0, k}] -2*B[n, k]*B[n+1, k]/(k+1); Table[T[n, k], {n, 0, 10}, {k, 0, n} ]//Flatten (* G. C. Greubel, Nov 23 2019 *)
PROG
(PARI) T(n, k) = b=binomial; 2 + sum(j=0, k, (-1)^j*b(n+1, j)*(2*(k-j)+1)^n) - 2*b(n, k)*b(n+1, k)/(k+1); \\ G. C. Greubel, Nov 23 2019
(Magma) B:=Binomial; [2*(1 - B(n, k)*B(n+1, k)/(k+1)) + (&+[(-1)^j*B(n+1, j) *(2*(k-j)+1)^n: j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 23 2019
(Sage) b=binomial; [[2 + sum( (-1)^j*b(n+1, j)*(2*(k-j)+1)^n for j in (0..k)) - 2*b(n, k)*b(n+1, k)/(k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 23 2019
(GAP) B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> 2*(1 - B(n, k)*B(n+1, k)/(k+1)) + Sum([0..k], j-> (-1)^j*B(n+1, j)*(2*(k-j)+1)^n) ))); # G. C. Greubel, Nov 23 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 14 2010
EXTENSIONS
Edited by G. C. Greubel, Nov 23 2019
STATUS
approved