A147564 Triangle, T(n, k) = coefficients [x^k]( p(x, n) ), where p(x, n) = (1+x)^n +2*x*[n=2] for n < 3, otherwise (1+x)^n + 2*(n*x+1)*(1+x)^(n-2) -2, read by rows.

1, 1, 1, 1, 4, 1, 1, 11, 9, 1, 1, 16, 24, 12, 1, 1, 21, 46, 42, 15, 1, 1, 26, 75, 100, 65, 18, 1, 1, 31, 111, 195, 185, 93, 21, 1, 1, 36, 154, 336, 420, 308, 126, 24, 1, 1, 41, 204, 532, 826, 798, 476, 164, 27, 1, 1, 46, 261, 792, 1470, 1764, 1386, 696, 207, 30, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = coefficients [x^k]( p(x, n) ), where p(x, n) = (1+x)^n + 2*(1+x)^(n-1) + 2*(n-1)*x*(1+x)^(n-2) for n > 2, p(x, 0) = 1, p(x, 1) = 1 + x, p(x, 2) = 1 + 4*x + x^2.

T(n, k) = [x^k]( p(x, n) ), where p(x, n) = (1+x)^n +2*x*[n=2] for n < 3, otherwise (1+x)^n + 2*(n*x+1)*(1+x)^(n-2) -2. - G. C. Greubel, Oct 27 2022

Example

Triangle begins as:
  1;
  1,  1;
  1,  4,   1;
  1, 11,   9,    1;
  1, 16,  24,   12,    1;
  1, 21,  46,   42,   15,    1;
  1, 26,  75,  100,   65,   18,    1;
  1, 31, 111,  195,  185,   93,   21,    1;
  1, 36, 154,  336,  420,  308,  126,   24,   1;
  1, 41, 204,  532,  826,  798,  476,  164,  27,   1;
  1, 46, 261,  792, 1470, 1764, 1386,  696, 207,  30,  1;
  1, 51, 325, 1125, 2430, 3486, 3402, 2250, 975, 255, 33,  1;

Mathematica

p[x_, n_]= (1+x)^n +If[n>=1, -2 +2*(1+x)^(n-1), 0] +If[n>2, 2*(n-1)*x*(1+x)^(n- 2), 0];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten

Prog

(Magma) // As a triangle
function p(n, x)
  if n le 1 then return (1+x)^n;
  elif n eq 2 then return 1 +4*x +x^2;
  else return (1+x)^n +2*(n*x+1)*(1+x)^(n-2) -2;
  end if; return p;
end function;
R<x>:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( p(n, x) )): n in [0..12]]; // G. C. Greubel, Oct 27 2022
(SageMath)
def p(n, x): return (1+x)^n +2*x*int(n==2) if (n<3) else (1+x)^n +2*(n*x+1)*(1+x)^(n-2) -2
flatten([[( p(n, x) ).series(x, n+1).list()[k] for k in range(n+1)] for n in (0..12)]) # G. C. Greubel, Oct 27 2022

Crossrefs

Sequence in context: A232774 A331969 A203860 * A090981 A087903 A287532

Adjacent sequences: A147561 A147562 A147563 * A147565 A147566 A147567

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 07 2008

Extensions

Edited by G. C. Greubel, Oct 27 2022

Status

approved