A381567
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 A381568.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 6, 14, 22, 0, 1, 8, 27, 64, 126, 0, 1, 10, 44, 134, 365, 884, 0, 1, 12, 65, 240, 777, 2492, 7149, 0, 1, 14, 90, 390, 1438, 5238, 19578, 64688, 0, 1, 16, 119, 592, 2440, 9696, 40244, 172356, 641836, 0, 1, 18, 152, 854, 3891, 16632, 73408, 345726, 1668686, 6888740, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, 12, ...
0, 5, 14, 27, 44, 65, 90, ...
0, 22, 64, 134, 240, 390, 592, ...
0, 126, 365, 777, 1438, 2440, 3891, ...
0, 884, 2492, 5238, 9696, 16632, 27036, ...
0, 7149, 19578, 40244, 73408, 125035, 203258, ...
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a(n, k) = 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)));
A143508
G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2)^2.
Original entry on oeis.org
1, 1, 2, 9, 52, 372, 3058, 28074, 282028, 3059328, 35497672, 437499541, 5696752234, 78036803430, 1120687989348, 16823652188164, 263345788211608, 4289062071449610, 72543038644585822, 1271980596430351862, 23085579883157411532, 433071407705851089244
Offset: 0
G.f. A(x) = 1 + x + 2*x^2 + 9*x^3 + 52*x^4 + 372*x^5 + 3058*x^6 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 126*x^4 + 884*x^5 + 7149*x^6 +...
A(x*A(x)^2) = 1 + x + 4*x^2 + 22*x^3 + 156*x^4 + 1285*x^5 + 11886*x^6 +...
A(x*A(x)^2)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 372*x^4 + 3058*x^5 +...
Define G(x) by G(x*A(x)^2) = x, then
G(x) = x - 2*x^2 + 3*x^3 - 12*x^4 + 17*x^5 - 198*x^6 - 345*x^7 +...
such that G(x) = x/(1 + A(x)^2*G(x))^2.
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{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^2,x,x*A^2));polcoeff(A,n)}
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Mar 01 2025
A381570
G.f. A(x) satisfies A(x) = (1 + x*A(x*A(x)))^3.
Original entry on oeis.org
1, 3, 12, 82, 732, 7944, 99156, 1381464, 21065853, 346932822, 6112226961, 114383442888, 2261347164766, 47025363829497, 1025005545866361, 23349137897005296, 554467427766694440, 13696046757037152183, 351231525904387758222, 9335221780768641038952
Offset: 0
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a(n, k=1) = 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)));
Showing 1-3 of 3 results.