A120917
Central terms of triangle A120914 (cascadence of (1+2x)^2).
Original entry on oeis.org
1, 4, 32, 212, 1504, 10848, 79696, 596160, 4520000, 34673940, 268538048, 2096374656, 16475970896, 130234435648, 1034568731408, 8254368150320, 66112337392256, 531345216883584, 4283682906179728, 34632004320564416
Offset: 0
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{a(n)=local(A,F=1+4*x+4*x^2,d=2,G=x,H=1+x,S=ceil(log(n+1)/log(d+1))); for(i=0,n,G=x*subst(F,x,G+x*O(x^n)));for(i=0,S,H=subst(H,x,x*G^d+x*O(x^n))*G/x); A=(x*H-y*subst(H,x,x*y^d +x*O(x^n)))/(x*subst(F,x,y)-y); polcoeff(polcoeff(A,n,x),n,y)}
A120918
Row sums of triangle A120914 (cascadence of (1+2x)^2).
Original entry on oeis.org
1, 12, 124, 1212, 11512, 107544, 994236, 9128024, 83400856, 759387964, 6896903064, 62519804504, 565914425336, 5116780986152, 46223426993576, 417279346904792, 3764890593799336, 33953608251139560, 306100904240342268
Offset: 0
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{a(n)=local(A,F=1+4*x+4*x^2,d=2,G=x,H=1+x,S=ceil(log(n+1)/log(d+1))); for(i=0,n,G=x*subst(F,x,G+x*O(x^n)));for(i=0,S,H=subst(H,x,x*G^d+x*O(x^n))*G/x); A=(x*H-y*subst(H,x,x*y^d +x*O(x^n)))/(x*subst(F,x,y)-y); sum(k=0,2*n,polcoeff(polcoeff(A,n,x),k,y))}
A120915
G.f. satisfies: A(x) = C(2x)^2 * A(x^3*C(2x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).
Original entry on oeis.org
1, 4, 20, 116, 720, 4656, 30996, 210896, 1459536, 10239796, 72651184, 520328112, 3756512912, 27307671040, 199705789248, 1468209751856, 10844681408064, 80437588353600, 598867568439828, 4473784063109904, 33524058847464912
Offset: 0
A(x) = 1 + 4*x + 20*x^2 + 116*x^3 + 720*x^4 + 4656*x^5 + 30996*x^6 +...
= C(2x)^2 * A(x^3*C(2x)^4) where
C(2x) = 1 + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1344*x^5 + 8448*x^6 +...
and C(x) is g.f. of the Catalan numbers (A000108): C(x) = 1 + x*C(x)^2.
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{a(n)=local(A=1+x,C=(1/x*serreverse(x/(1+4*x+4*x^2+x*O(x^n))))^(1/2)); for(i=0,n,A=C^2*subst(A,x,x^3*C^4 +x*O(x^n)));polcoeff(A,n,x)}
A120916
G.f. satisfies: A(x) = C(2x)*A(x^3*C(2x)^4), where C(x) is the g.f. of the Catalan numbers (A000108).
Original entry on oeis.org
1, 2, 8, 42, 244, 1504, 9656, 63856, 431872, 2972778, 20756036, 146627648, 1046060836, 7525452296, 54530660832, 397628393728, 2915496099136, 21481907631872, 158975372309176, 1181109256858096, 8806197969093184
Offset: 0
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{a(n)=local(A=1+2*x,C=(1/x*serreverse(x/(1+4*x+4*x^2+x*O(x^n))))^(1/2)); for(i=0,n,A=C*subst(A,x,x^3*C^4 +x*O(x^n)));polcoeff(A,n,x)}
Showing 1-4 of 4 results.
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