%I #31 Jun 11 2022 11:42:19

%S 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,

%T 6,1,1023,2062,1767,861,265,57,7,1,4926,10610,9873,5300,1845,440,77,8,

%U 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

%N 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.

%H Paul D. Hanna, <a href="/A325580/b325580.txt">Table of n, a(n) for n = 0..5150 terms of this triangle as read by rows 0..100</a>

%F G.f.: A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k equals the following.

%F (1) A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n.

%F (2) A(x,y) = Sum_{n>=0} x^n * (1+x)^(n^2) / (1 - x*y*(1+x)^n)^(n+1).

%F (3) A(x,y) = Sum_{k>=0} y^k * Sum_{n>=0} binomial(n+k,n) * (x*(1+x)^n)^(n+k).

%F G.f. of column k: Sum_{n>=0} binomial(n+k,n) * x^n * (1+x)^(n*(n+k)).

%e G.f. A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k begins:

%e 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 + ...

%e where, by definition,

%e A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n.

%e This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 5, 7, 3, 1;

%e 16, 24, 15, 4, 1;

%e 57, 98, 67, 26, 5, 1;

%e 231, 430, 336, 144, 40, 6, 1;

%e 1023, 2062, 1767, 861, 265, 57, 7, 1;

%e 4926, 10610, 9873, 5300, 1845, 440, 77, 8, 1;

%e 25483, 58240, 58221, 33974, 13041, 3501, 679, 100, 9, 1;

%e 140601, 338984, 360930, 226716, 94580, 27978, 6083, 992, 126, 10, 1;

%e 822422, 2081189, 2345469, 1572134, 706225, 226843, 54271, 9886, 1389, 155, 11, 1;

%e 5074015, 13423258, 15926115, 11318196, 5428820, 1876728, 486941, 97448, 15246, 1880, 187, 12, 1; ...

%e the leftmost column in which yields A121689:

%e [1, 1, 2, 5, 16, 57, 231, 1023, 4926, 25483, 140601, ..., A121689, ...]

%e and has g.f.: Sum_{n>=0} x^n * (1+x)^(n^2).

%e Column 1 equals

%e [1, 2, 7, 24, 98, 430, 2062, 10610, 58240, 338984, ..., A325581(n), ...]

%e and has g.f.: Sum_{n>=0} (n+1) * x^n * (1+x)^(n*(n+1)).

%e Column 2 equals

%e [1, 3, 15, 67, 336, 1767, 9873, 58221, 360930, ..., A325586(n), ...]

%e and has g.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1+x)^(n*(n+2)).

%e The row sums of this triangle begin

%e [1, 2, 5, 16, 60, 254, 1188, 6043, 33080, 193249, ..., A301306(n), ...]

%e and has g.f.: Sum_{n>=0} (1 + (1+x)^n)^n * x^n.

%o (PARI) {T(n,k) = my(Axy = sum(m=0,n, x^m * ((1+x +x*O(x^n))^m + y)^m ) );

%o polcoeff( polcoeff( Axy,n,x),k,y)}

%o for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))

%Y Cf. A121689 (column 0), A301306 (row sums), A325581 (column 1), A325586 (column 2), A325587 (column 3).

%K sign,tabl

%O 0,4

%A _Paul D. Hanna_, May 11 2019