A381569
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381570.
Original entry on oeis.org
1, 1, 0, 1, 3, 0, 1, 6, 12, 0, 1, 9, 33, 82, 0, 1, 12, 63, 236, 732, 0, 1, 15, 102, 489, 2100, 7944, 0, 1, 18, 150, 868, 4428, 22248, 99156, 0, 1, 21, 207, 1400, 8121, 46422, 270268, 1381464, 0, 1, 24, 273, 2112, 13665, 85272, 552540, 3668568, 21065853, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 3, 6, 9, 12, 15, ...
0, 12, 33, 63, 102, 150, ...
0, 82, 236, 489, 868, 1400, ...
0, 732, 2100, 4428, 8121, 13665, ...
0, 7944, 22248, 46422, 85272, 145143, ...
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a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+3*k, j)/(n-j+k)*a(n-j, j)));
A212029
G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3)^3.
Original entry on oeis.org
1, 1, 3, 21, 190, 2112, 26922, 382110, 5920788, 98862273, 1762572957, 33325846461, 664774457583, 13932829786025, 305788481726799, 7008171327166869, 167321925537782445, 4153009604547937170, 106963758805117459392, 2854029374011293902121, 78773444214057182702790
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 190*x^4 + 2112*x^5 + 26922*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 12*x^2 + 82*x^3 + 732*x^4 + 7944*x^5 + 99156*x^6 +..
A(x*A(x)^3) = 1 + x + 6*x^2 + 51*x^3 + 560*x^4 + 7155*x^5 + 102495*x^6 +...
A(x*A(x)^3)^3 = 1 + 3*x + 21*x^2 + 190*x^3 + 2112*x^4 + 26922*x^5 +...
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{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, x, x*A^3)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, 3*j))); \\ Seiichi Manyama, Mar 01 2025
A381574
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x*A(x)))^3.
Original entry on oeis.org
1, 3, 15, 118, 1206, 14712, 204385, 3143826, 52580328, 944416084, 18056415144, 365065244238, 7765839784508, 173123253590079, 4031536347783786, 97807655876704029, 2466489368705170539, 64527021089110890192, 1748298996924574135699, 48982266056400514509660
Offset: 0
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a(n, k=1) = if(k==0, 0^n, 3*k*sum(j=0, n, binomial(3*n-2*j+3*k, j)/(3*n-2*j+3*k)*a(n-j, j)));
A381568
G.f. A(x) satisfies A(x) = (1 + x*A(x*A(x)))^2.
Original entry on oeis.org
1, 2, 5, 22, 126, 884, 7149, 64688, 641836, 6888740, 79203860, 968503090, 12525131474, 170555767116, 2436592516874, 36409825487380, 567612675812796, 9211031425896752, 155283809480528788, 2714788300934206360, 49140787009610861896, 919625415852055598804, 17768937720619971300781
Offset: 0
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+2*k, j)/(n-j+k)*a(n-j, j)));
Showing 1-4 of 4 results.