A154869 A triangular sequence: T(n,m) = t1(n,m) + t1(n,n-m) where t1(n,m) = -Sum_{j=0..m+1} (-1)^j * t0(n + 2, j) * (m - j + 1)^(n + 1) and t0(n,m) = Sum_{j=0..m+1} (-1)^j * binomial(n + 2, j) * (m - j + 1)^(n + 1).

6, 26, 26, 230, 100, 230, 3092, 857, 857, 3092, 53032, 13671, 4816, 13671, 53032, 1094774, 285588, 64514, 64514, 285588, 1094774, 26402826, 7001142, 1517286, 474132, 1517286, 7001142, 26402826, 728697032, 195578147, 43758387, 8678237

Offset

0,1

Comments

Row sums are: 6, 52, 560, 7898, 138222, 2889752, 70316640, 1953423606, 61038674510, 2119955154436, 81049092159048, ...

The t0 numbers in the name are the Eulerian triangle A008292. - Sean A. Irvine, Mar 01 2020

Links

Table of n, a(n) for n=0..31.

Formula

t0(n,m) = Sum_{j=0..m+1} (-1)^j*binomial(n + 2, j) * (m - j + 1)^(n + 1);

t1(n,m) = -Sum_{j=0..m+1} (-1)^j* t0(n + 2, j) * (m - j + 1)^(n + 1);

T(n,m) = t1(n,m) + t1(n,n-m).

Equivalently, t1(n,m) = (1/(n+4)) * Sum_{j=0..m} (j+2) * binomial(n+4,j+2) * (m-j+1)^(n+1). - Sean A. Irvine, Mar 03 2020

Example

Triangle begins:
  6;
  26, 26;
  230, 100, 230;
  3092, 857, 857, 3092;
  53032, 13671, 4816, 13671, 53032;
  1094774, 285588, 64514, 64514, 285588, 1094774;
  26402826, 7001142, 1517286, 474132, 1517286, 7001142, 26402826;
  728697032, 195578147, 43758387, 8678237, 8678237, 43758387, 195578147, 728697032;

Maple

t0 := proc(n, m) option remember;
    sum(((-1)^j)*binomial(n + 2, j)*(m - j + 1)^(n + 1), j = 0..m+1)
end proc:
t := proc(n, m) option remember;
    - sum(((-1)^j)*t0(n + 2, j)*(m - j + 1)^(n + 1), j = 0..m+1)
end proc:
seq(seq(t(n, m) + t(n, n - m), m = 0..n), n=0..10);
# Yu-Sheng Chang and Georg Fischer, Feb 03 2020

Mathematica

t0[n_, m_] := Sum[(-1)^j*Binomial[n+2, j]*(-j + m + 1)^(n+1), {j, 0, m+1}];
t[n_, m_] := -Sum[((-1)^(2*j + 1)*(j + 2)*Binomial[n + 4, j + 2]*(-j + m + 1)^(n + 1))/(n + 4), {j, 0, m + 1}];
Table[Table[t[n, n - m] + t[n, m], {m, 0, n}], {n, 0, 10}] // Flatten (* edited by Jean-François Alcover, Mar 14 2020 *)

Crossrefs

Sequence in context: A337400 A036175 A239178 * A043354 A023727 A045255

Adjacent sequences: A154866 A154867 A154868 * A154870 A154871 A154872

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 16 2009

Status

approved