A227373 Antidiagonal sums of triangle A227372.
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
Keywords
Examples
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.
Programs
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PARI
/* 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),", "))
Formula
G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2*B(x), where B(x) = 1 + x^2*B(x)^2*C(x), C(x) = 1 + x^3*C(x)^2*D(x), D(x) = 1 + x^4*D(x)^2*E(x), etc.
Comments