A227372
G.f.: A(x,q) = 1 + x*A(q*x,q) * A(x,q)^2.
Original entry on oeis.org
1, 1, 2, 1, 5, 4, 2, 1, 14, 15, 10, 9, 4, 2, 1, 42, 56, 45, 43, 34, 23, 14, 9, 4, 2, 1, 132, 210, 196, 196, 174, 156, 121, 85, 59, 42, 27, 14, 9, 4, 2, 1, 429, 792, 840, 882, 842, 796, 749, 627, 480, 382, 289, 216, 157, 101, 67, 46, 27, 14, 9, 4, 2, 1, 1430, 3003
Offset: 0
Triangle begins:
[1];
[1];
[2, 1];
[5, 4, 2, 1];
[14, 15, 10, 9, 4, 2, 1];
[42, 56, 45, 43, 34, 23, 14, 9, 4, 2, 1];
[132, 210, 196, 196, 174, 156, 121, 85, 59, 42, 27, 14, 9, 4, 2, 1];
[429, 792, 840, 882, 842, 796, 749, 627, 480, 382, 289, 216, 157, 101, 67, 46, 27, 14, 9, 4, 2, 1];
[1430, 3003, 3564, 3942, 3990, 3921, 3848, 3681, 3242, 2732, 2267, 1838, 1489, 1189, 909, 671, 494, 345, 252, 173, 109, 71, 46, 27, 14, 9, 4, 2, 1]; ...
Explicitly, the polynomials in q begin:
1;
1;
2 + q;
5 + 4*q + 2*q^2 + q^3;
14 + 15*q + 10*q^2 + 9*q^3 + 4*q^4 + 2*q^5 + q^6;
42 + 56*q + 45*q^2 + 43*q^3 + 34*q^4 + 23*q^5 + 14*q^6 + 9*q^7 + 4*q^8 + 2*q^9 + q^10;
132 + 210*q + 196*q^2 + 196*q^3 + 174*q^4 + 156*q^5 + 121*q^6 + 85*q^7 + 59*q^8 + 42*q^9 + 27*q^10 + 14*q^11 + 9*q^12 + 4*q^13 + 2*q^14 + q^15; ...
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{T(n,k)=local(A=1);for(i=1,n,A=1+x*subst(A,x,q*x)*A^2 +x*O(x^n));polcoeff(polcoeff(A,n,x),k,q)}
for(n=0,10,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))
A227373
Antidiagonal sums of triangle A227372.
Original entry on oeis.org
1, 1, 2, 6, 18, 59, 199, 693, 2465, 8937, 32880, 122513, 461331, 1753037, 6713758, 25888515, 100427611, 391657635, 1534674930, 6039078032, 23855475724, 94561195899, 376019415794, 1499554893338, 5996061250461, 24034238674758, 96554979145357, 388711331661818, 1567919554600690
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 59*x^5 + 199*x^6 + 693*x^7 +...
and equals a series involving row polynomials of triangle A227372:
A(x) = 1 + x*(1) + x^2*(2 + x) + x^3*(5 + 4*x + 2*x^2 + x^3)
+ x^4*(14 + 15*x + 10*x^2 + 9*x^3 + 4*x^4 + 2*x^5 + x^6)
+ x^5*(42 + 56*x + 45*x^2 + 43*x^3 + 34*x^4 + 23*x^5 + 14*x^6 + 9*x^7 + 4*x^8 + 2*x^9 + x^10) +...
RELATED SERIES.
G.f. A(x) = 1 + x*A(x)^2*B(x), where
B(x) = 1 + x^2 + 2*x^4 + x^5 + 5*x^6 + 4*x^7 + 16*x^8 + 16*x^9 + 52*x^10 +...
and B(x) = 1 + x^2*B(x)^2*C(x), where
C(x) = 1 + x^3 + 2*x^6 + x^7 + 5*x^9 + 4*x^10 + 2*x^11 + 15*x^12 +...
and C(x) = 1 + x^3*C(x)^2*D(x), where
D(x) = 1 + x^4 + 2*x^8 + x^9 + 5*x^12 + 4*x^13 + 2*x^14 + x^15 + 14*x^16 +...
and D(x) = 1 + x^4*D(x)^2*E(x), where
E(x) = 1 + x^5 + 2*x^10 + x^11 + 5*x^15 + 4*x^16 + 2*x^17 + x^18 + 14*x^20 +...
etc.
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/* From g.f. of A227372: G(x,q) = 1 + x*G(q*x,q)*G(x,q)^2: */
{a(n)=local(G=1);for(i=1,n,G=1+x*subst(G,x,q*x)*G^2 +x*O(x^n));polcoeff(sum(m=0,n,q^m*polcoeff(G,m,x))+q*O(q^n),n,q)}
for(n=0,40,print1(a(n),", "))
Showing 1-2 of 2 results.
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