A203148 -id:A203148 - cp's OEIS frontend

A173007 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n and q = 3.

1, 3, 1, 27, 12, 1, 729, 351, 39, 1, 59049, 29160, 3510, 120, 1, 14348907, 7144929, 882090, 32670, 363, 1, 10460353203, 5223002148, 650188539, 24698520, 297297, 1092, 1, 22876792454961, 11433166050879, 1427185336941, 54665851779, 674887059, 2685501, 3279, 1

Offset: 0

Roger L. Bagula, Feb 07 2010

Triangle T(n,k), read by rows, given by [3,6,27,72,243,702,2187,6480,...] DELTA [1,0,3,0,9,0,27,0,81,0,243,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 01 2011

			Triangle begins as:
            1;
            3,          1;
           27,         12,         1;
          729,        351,        39,        1;
        59049,      29160,      3510,      120,      1;
     14348907,    7144929,    882090,    32670,    363,    1;
  10460353203, 5223002148, 650188539, 24698520, 297297, 1092, 1;

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

Cf. A023531 (q=0), A007318 (q=1), A108084 (q=2), this sequence (q=3), A173008 (q=4).

Cf. A203148, A290000.

function T(n,k,q)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
  end if; return T; end function;
[T(n,k,3): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

(* First program *)
p[x_, n_, q_] = If[n==0, 1, Product[x + q^i, {i, 1, n}]];
Table[CoefficientList[p[x, n, 3], x], {n, 0, 10}] (* modified by G. C. Greubel, Feb 20 2021 *)
(* Second program *)
T[n_, k_, q_]:= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1,k,q] +T[n-1,k-1,q] ]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

def T(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

p(x,n,q) = 1 if n=0, Product_{i=1..n} (x + q^i) otherwise, with q=3.
T(n,k) = 3^n*T(n-1,k) + T(n-1,k-1), T(0,0)=1. - Philippe Deléham, Oct 01 2011
Sum_{k=0..n} T(n, k, 4) = A290000(n+1). - G. C. Greubel, Feb 20 2021

Edited by G. C. Greubel, Feb 20 2021

A203229 (n-1)-st elementary symmetric function of (1,16,...,n^4).

Original entry on oeis.org

1, 17, 1393, 357904, 224021776, 290539581696, 697854274212096, 2859056348455305216, 18760911610508623282176, 187626456226399005573120000, 2747212346823835568109649920000, 56968733990900457398848318341120000

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A000583, A203148.

Column k=4 of A291556.

f[k_] := k^4; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 14}]     (* A203229 *)
Table[(n!)^4 * Sum[1/i^4, {i, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(n) ~ 2 * Pi^6 * n^(4*n+2) / (45*exp(4*n)). - Vaclav Kotesovec, Aug 27 2017

Sum_{n>=1} a(n) * x^n / (n!)^4 = polylog(4,x) / (1 - x). - Ilya Gutkovskiy, Jul 14 2020

A203149 (n-1)-st elementary symmetric function of {2,8,26,80,242,...,-1+3^n}.

Original entry on oeis.org

1, 10, 276, 22496, 5477312, 3995536896, 8740106791936, 57347917373194240, 1128805788065906196480, 66655379003341682687344640, 11807831483305724934163060490240, 6275171273199511284527725270165094400, 10004652813703079731923092657993740504268800

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

Cf. A203148.

f[k_] := 3^k - 1; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203149 *)

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A173007 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n and q = 3.

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A203229 (n-1)-st elementary symmetric function of (1,16,...,n^4).

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Mathematica

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A203149 (n-1)-st elementary symmetric function of {2,8,26,80,242,...,-1+3^n}.

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Mathematica