A155917 A difference triangle of Pascal-Sierpinski 5th level and the Pascal second derivative: a(n,k)= (4*n - 4*k + 1)a(n - 1, k - 1) + (4*k - 3)a(n - 1, k); p(x,n)=(Sum[10*n*(n - 1)*a(n, k)*x^(k - 1) - D[(x + 1)^(n + 2), {x, 2}]/(x + 1), {k, n}])/2.

-3, -2, -2, 0, 240, 0, 3360, 3360, -5, 30380, 105570, 30380, -5, -18, 232710, 2032620, 2032620, 232710, -18, -42, 1637748, 31186890, 74043480, 31186890, 1637748, -42, -80, 10932880, 420179760, 1990483600, 1990483600, 420179760, 10932880, -80

Offset: 1

Author

Roger L. Bagula, Jan 30 2009

Keywords

• sign
• tabl
• uned

Comments

Row sums are:

Examples

			{-3},
{-2, -2},
{0, 240},
{0, 3360, 3360},
{-5, 30380, 105570, 30380, -5},
{-18, 232710, 2032620, 2032620, 232710, -18},
{-42, 1637748, 31186890, 74043480, 31186890, 1637748, -42},
{-80, 10932880, 420179760, 1990483600, 1990483600, 420179760, 10932880, -80},
{-135, 70305480, 5213648700, 44614752120, 87013084950, 44614752120, 5213648700, 70305480, -135},
{-210, 439442910, 61202397240, 887917071960, 3020166679140, 3020166679140, 887917071960, 61202397240, 439442910, -210}
		

Crossrefs

A142459

Programs

Mathematica

A[n_, 1] := 1; A[n_, n_] := 1;
A[n_, k_] := (4*n - 4*k + 1)A[n - 1, k - 1] + (4*k - 3)A[n - 1, k];
a = Table[ExpandAll[(Sum[10*n*(n - 1)*A[n, k]*x^(k - 1) - D[(x + 1)^(n + 2), {x, 2}]/(x + 1), {k, n}])/2], {n, 10}];
Table[CoefficientList[ExpandAll[a[[n]]], x], {n, 1, Length[a]}];
Flatten[%]

Formula

a(n,k)= (4*n - 4*k + 1)a(n - 1, k - 1) + (4*k - 3)a(n - 1, k);

p(x,n)=(Sum[10*n*(n - 1)*a(n, k)*x^(k - 1) - D[(x + 1)^(n + 2), {x, 2}]/(x + 1), {k, n}])/2;

t(n,m)=coefficients(p(x,n)).