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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.
4
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
0,2
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
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).
Sequence in context: A368266 A338681 A069740 * A298828 A114917 A100864
KEYWORD
nonn,tabl,easy
AUTHOR
Roger L. Bagula, Feb 07 2010
STATUS
approved