A325580 G.f.: A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n, where A(0) = 0, as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k, read by rows.
1, 1, 1, 2, 2, 1, 5, 7, 3, 1, 16, 24, 15, 4, 1, 57, 98, 67, 26, 5, 1, 231, 430, 336, 144, 40, 6, 1, 1023, 2062, 1767, 861, 265, 57, 7, 1, 4926, 10610, 9873, 5300, 1845, 440, 77, 8, 1, 25483, 58240, 58221, 33974, 13041, 3501, 679, 100, 9, 1, 140601, 338984, 360930, 226716, 94580, 27978, 6083, 992, 126, 10, 1, 822422, 2081189, 2345469, 1572134, 706225, 226843, 54271, 9886, 1389, 155, 11, 1, 5074015, 13423258, 15926115, 11318196, 5428820, 1876728, 486941, 97448, 15246, 1880, 187, 12, 1
Offset: 0
Examples
G.f. A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k begins:
A(x,y) = 1 + (y + 1)*x + (y^2 + 2*y + 2)*x^2 + (y^3 + 3*y^2 + 7*y + 5)*x^3 + (y^4 + 4*y^3 + 15*y^2 + 24*y + 16)*x^4 + (y^5 + 5*y^4 + 26*y^3 + 67*y^2 + 98*y + 57)*x^5 + (y^6 + 6*y^5 + 40*y^4 + 144*y^3 + 336*y^2 + 430*y + 231)*x^6 + (y^7 + 7*y^6 + 57*y^5 + 265*y^4 + 861*y^3 + 1767*y^2 + 2062*y + 1023)*x^7 + (y^8 + 8*y^7 + 77*y^6 + 440*y^5 + 1845*y^4 + 5300*y^3 + 9873*y^2 + 10610*y + 4926)*x^8 + (y^9 + 9*y^8 + 100*y^7 + 679*y^6 + 3501*y^5 + 13041*y^4 + 33974*y^3 + 58221*y^2 + 58240*y + 25483)*x^9 + (y^10 + 10*y^9 + 126*y^8 + 992*y^7 + 6083*y^6 + 27978*y^5 + 94580*y^4 + 226716*y^3 + 360930*y^2 + 338984*y + 140601)*x^10 + ...
where, by definition,
A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n.
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins
1;
1, 1;
2, 2, 1;
5, 7, 3, 1;
16, 24, 15, 4, 1;
57, 98, 67, 26, 5, 1;
231, 430, 336, 144, 40, 6, 1;
1023, 2062, 1767, 861, 265, 57, 7, 1;
4926, 10610, 9873, 5300, 1845, 440, 77, 8, 1;
25483, 58240, 58221, 33974, 13041, 3501, 679, 100, 9, 1;
140601, 338984, 360930, 226716, 94580, 27978, 6083, 992, 126, 10, 1;
822422, 2081189, 2345469, 1572134, 706225, 226843, 54271, 9886, 1389, 155, 11, 1;
5074015, 13423258, 15926115, 11318196, 5428820, 1876728, 486941, 97448, 15246, 1880, 187, 12, 1; ...
the leftmost column in which yields A121689:
[1, 1, 2, 5, 16, 57, 231, 1023, 4926, 25483, 140601, ..., A121689, ...]
and has g.f.: Sum_{n>=0} x^n * (1+x)^(n^2).
Column 1 equals
[1, 2, 7, 24, 98, 430, 2062, 10610, 58240, 338984, ..., A325581(n), ...]
and has g.f.: Sum_{n>=0} (n+1) * x^n * (1+x)^(n*(n+1)).
Column 2 equals
[1, 3, 15, 67, 336, 1767, 9873, 58221, 360930, ..., A325586(n), ...]
and has g.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1+x)^(n*(n+2)).
The row sums of this triangle begin
[1, 2, 5, 16, 60, 254, 1188, 6043, 33080, 193249, ..., A301306(n), ...]
and has g.f.: Sum_{n>=0} (1 + (1+x)^n)^n * x^n.
Links
Crossrefs
Programs
-
PARI
{T(n,k) = my(Axy = sum(m=0,n, x^m * ((1+x +x*O(x^n))^m + y)^m ) ); polcoeff( polcoeff( Axy,n,x),k,y)} for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))
Formula
G.f.: A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k equals the following.
(1) A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n.
(2) A(x,y) = Sum_{n>=0} x^n * (1+x)^(n^2) / (1 - x*y*(1+x)^n)^(n+1).
(3) A(x,y) = Sum_{k>=0} y^k * Sum_{n>=0} binomial(n+k,n) * (x*(1+x)^n)^(n+k).
G.f. of column k: Sum_{n>=0} binomial(n+k,n) * x^n * (1+x)^(n*(n+k)).
Comments