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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.

%I A155917 #4 Jun 02 2025 01:19:47
%S A155917 -3,-2,-2,0,240,0,3360,3360,-5,30380,105570,30380,-5,-18,232710,
%T A155917 2032620,2032620,232710,-18,-42,1637748,31186890,74043480,31186890,
%U A155917 1637748,-42,-80,10932880,420179760,1990483600,1990483600,420179760,10932880,-80
%N 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.
%C A155917 Row sums are:
%F A155917 a(n,k)= (4*n - 4*k + 1)a(n - 1, k - 1) + (4*k - 3)a(n - 1, k);
%F A155917 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;
%F A155917 t(n,m)=coefficients(p(x,n)).
%e A155917 {-3},
%e A155917 {-2, -2},
%e A155917 {0, 240},
%e A155917 {0, 3360, 3360},
%e A155917 {-5, 30380, 105570, 30380, -5},
%e A155917 {-18, 232710, 2032620, 2032620, 232710, -18},
%e A155917 {-42, 1637748, 31186890, 74043480, 31186890, 1637748, -42},
%e A155917 {-80, 10932880, 420179760, 1990483600, 1990483600, 420179760, 10932880, -80},
%e A155917 {-135, 70305480, 5213648700, 44614752120, 87013084950, 44614752120, 5213648700, 70305480, -135},
%e A155917 {-210, 439442910, 61202397240, 887917071960, 3020166679140, 3020166679140, 887917071960, 61202397240, 439442910, -210}
%t A155917 A[n_, 1] := 1; A[n_, n_] := 1;
%t A155917 A[n_, k_] := (4*n - 4*k + 1)A[n - 1, k - 1] + (4*k - 3)A[n - 1, k];
%t A155917 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}];
%t A155917 Table[CoefficientList[ExpandAll[a[[n]]], x], {n, 1, Length[a]}];
%t A155917 Flatten[%]
%Y A155917 A142459
%K A155917 sign,tabl,uned
%O A155917 1,1
%A A155917 _Roger L. Bagula_, Jan 30 2009