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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

Robert Israel, Table of n, a(n) for n = 0..1430

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

EXAMPLE

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;

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 *)

PROG

(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

(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

CROSSREFS

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

Cf. A092896, A108084, A309327.

Sequence in context: A113112 A278578 A069740 * A298828 A114917 A100864

Adjacent sequences:  A173005 A173006 A173007 * A173009 A173010 A173011

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Feb 07 2010

STATUS

approved

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Last modified April 21 07:26 EDT 2021. Contains 343146 sequences. (Running on oeis4.)