A243950 Sum of the squares of q-binomial coefficients for q=2 in row n of triangle A022166, for n >= 0.
1, 2, 11, 100, 1677, 49974, 2801567, 293257480, 59426801521, 23154622451162, 17786849024835651, 26694462878992491180, 79786045619298591331605, 469805503062346255040726910, 5538428985758278544518994721255, 129179377104085570277109465712798800, 6048537751321912538368011648067930447545
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 11*x^2 + 100*x^3 + 1677*x^4 + 49974*x^5 + 2801567*x^6 + ... Related integer series: A(x)^(1/2) = 1 + x + 5*x^2 + 45*x^3 + 781*x^4 + 23981*x^5 + 1371885*x^6 + 145101805*x^7 + 29560055405*x^8 + ... + A243951(n)*x^n + ... A022166, the triangle of q-binomial coefficients for q=2, begins: 1; 1, 1; 1, 3, 1; 1, 7, 7, 1; 1, 15, 35, 15, 1; 1, 31, 155, 155, 31, 1; 1, 63, 651, 1395, 651, 63, 1; 1, 127, 2667, 11811, 11811, 2667, 127, 1; ... from which we can illustrate the initial terms of this sequence: a(0) = 1^2 = 1; a(1) = 1^2 + 1^2 = 2; a(2) = 1^2 + 3^2 + 1^2 = 11; a(3) = 1^2 + 7^2 + 7^2 + 1^2 = 100; a(4) = 1^2 + 15^2 + 35^2 + 15^2 + 1^2 = 1677; a(5) = 1^2 + 31^2 + 155^2 + 155^2 + 31^2 + 1^2 = 49974; a(6) = 1^2 + 63^2 + 651^2 + 1395^2 + 651^2 + 63^2 + 1^2 = 2801567; ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..60
- Wikipedia, Green's relations
Programs
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Mathematica
a[n_] := Sum[QBinomial[n, k, 2]^2, {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *) -
PARI
{A022166(n, k)=polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)} {a(n)=sum(k=0,n,A022166(n, k)^2)} for(n=0,20,print1(a(n),", "))
Formula
a(n) ~ c * 2^(n^2/2), where c = 18.0796893855819714431... if n is even and c = 18.02126069886312898683... if n is odd. - Vaclav Kotesovec, Jun 23 2014
Sum_{n>=0} a(n)*x^n/A005329(n)^2 = E(x)^2 where E(x) = Sum_{n>=0} x^n/A005329(n)^2. - Geoffrey Critzer, Jun 20 2023
Comments