cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

Roger L. Bagula, Feb 07 2010

Keywords

Comments

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

Examples

			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;

Links

Crossrefs

Cf. A023531 (q=0), A007318 (q=1), A108084 (q=2), this sequence (q=3), A173008 (q=4).
Cf. A203148, A290000.

Programs

  • Magma
    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
  • Mathematica
    (* 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 *)
  • Sage
    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

Formula

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

Extensions

Edited by G. C. Greubel, Feb 20 2021

A173008 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n, column 0<=k<=n, and q = 4.

Original entry on oeis.org

1, 4, 1, 64, 20, 1, 4096, 1344, 84, 1, 1048576, 348160, 22848, 340, 1, 1073741824, 357564416, 23744512, 371008, 1364, 1, 4398046511104, 1465657589760, 97615085568, 1543393280, 5957952, 5460, 1, 72057594037927936, 24017731997138944, 1600791219535872, 25384570585088, 99158478848, 95414592, 21844, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 07 2010

Keywords

Comments

Row sums are 1, 5, 85, 5525, 1419925, 1455423125, 5962868543125, 97701601079103125, 6403069829921181503125, ... (partial products of A092896).
Triangle T(n,k), read by rows, given by [4,12,64,240,1024,4032,16384,...] DELTA [1,0,4,0,16,0,64,0,256,0,1024,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 01 2011

Examples

			Triangle begins as:
              1;
              4,             1;
             64,            20,           1;
           4096,          1344,          84,          1;
        1048576,        348160,       22848,        340,       1;
     1073741824,     357564416,    23744512,     371008,    1364,    1;
  4398046511104, 1465657589760, 97615085568, 1543393280, 5957952, 5460, 1;

Links

Crossrefs

Cf. A023531 (q=0), A007318 (q=1), A108084 (q=2), A173007 (q=3), this sequence (q=4).
Cf. A092896, A108084, A309327.

Programs

  • Magma
    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,4): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021
  • Maple
    P:= 1: A:= 1:
for n from 1 to 12 do
  P:= expand(P*(x+4^n));
  A:= A, seq(coeff(P,x,j),j=0..n)
od:
A; # Robert Israel, Aug 12 2015
  • Mathematica
    (* First program *)
p[x_, n_, q_]= If[n==0, 1, Product[x + q^i, {i,n}]];
Table[CoefficientList[p[x, n, 4], x], {n, 0, 10}]//Flatten (* 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,4], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)
  • Sage
    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,4) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

Formula

T(n,k) = 4^n*T(n-1,k) + T(n-1,k-1) with T(0,0)=1. - Philippe Deléham, Oct 01 2011
Sum_{k=0..n} T(n, k, 4) = A309327(n+1). - G. C. Greubel, Feb 20 2021

A348014 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^k*x).

Original entry on oeis.org

1, 1, 1, 1, 5, 4, 1, 32, 139, 108, 1, 288, 8331, 35692, 27648, 1, 3413, 908331, 26070067, 111565148, 86400000, 1, 50069, 160145259, 42405161203, 1216436611100, 5205269945088, 4031078400000
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2021

Keywords

Examples

			Triangle begins:
  1;
  1,    1;
  1,    5,      4;
  1,   32,    139,      108;
  1,  288,   8331,    35692,     27648;
  1, 3413, 908331, 26070067, 111565148, 86400000;

Links

Crossrefs

Column k=1 gives A001923.
The diagonal of the triangle is A002109.
Cf. A007318, A008955, A023531, A108084, A173007, A173008, A249677.

Programs

  • PARI
    T(n, k) = if(k==0, 1, if(k==n, prod(j=1, n, j^j), T(n-1, k)+n^n*T(n-1, k-1)));
  • PARI
    row(n) = Vecrev(prod(k=1, n, 1+k^k*x));

Formula

T(0,0) = 1; T(n,k) = T(n-1,k) + n^n * T(n-1,k-1).
Showing 1-3 of 3 results.

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