A384580
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 A143500.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 8, 10, 0, 1, 5, 10, 16, 27, 46, 0, 1, 6, 15, 28, 54, 118, 244, 0, 1, 7, 21, 45, 95, 228, 609, 1481, 0, 1, 8, 28, 68, 155, 392, 1144, 3602, 10020, 0, 1, 9, 36, 98, 240, 631, 1916, 6597, 23866, 74400, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 3, 6, 10, 15, 21, ...
0, 3, 8, 16, 28, 45, 68, ...
0, 10, 27, 54, 95, 155, 240, ...
0, 46, 118, 228, 392, 631, 972, ...
0, 244, 609, 1144, 1916, 3015, 4560, ...
-
a(n, k) = 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, j)));
A143501
G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3).
Original entry on oeis.org
1, 1, 1, 4, 16, 92, 616, 4729, 40776, 388057, 4028230, 45207583, 544680014, 7004865885, 95694153485, 1382946630490, 21067128029388, 337224872043659, 5656357906530796, 99168643108816180, 1813250965008114981, 34506927801196386939, 682210688659107549765, 13988252589131863518730
Offset: 0
G.f. A(x) = 1 + x + x^2 + 4*x^3 + 16*x^4 + 92*x^5 + 616*x^6 + 4729*x^7 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 19*x^3 + 78*x^4 + 411*x^5 + 2617*x^6 +...
A(x*A(x)^3) = 1 + x + 4*x^2 + 16*x^3 + 92*x^4 + 616*x^5 + 4729*x^6 +...
If G(x*A(x)^3) = x then
G(x) = x - 3*x^2 + 12*x^3 - 64*x^4 + 372*x^5 - 2385*x^6 + 15675*x^7 -+...
A(G(x)) = 1 + A(x)*G(x) = (x/G(x))^(1/3) where
A(x)*G(x) = x - 2*x^2 + 10*x^3 - 51*x^4 + 324*x^5 - 1985*x^6 + 13938*x^7 -...
-
{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,x*A^3));polcoeff(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, j))); \\ Seiichi Manyama, Jun 04 2025
A143426
G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^2.
Original entry on oeis.org
1, 1, 2, 7, 32, 175, 1086, 7429, 54994, 435120, 3647686, 32192596, 297654824, 2872372828, 28841766844, 300592170551, 3244942353856, 36219458512421, 417365572999944, 4958429472475171, 60659660219655616, 763325035692109389, 9870492111677035538
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 175*x^5 + 1086*x^6 +...
A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 434*x^5 + 2986*x^6 +...
A(x*A(x))^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 175*x^4 + 1086*x^5 +...
Logarithmic series:
log(A(x)) = x + [d/dx x^3*A(x)^4]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^6]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^8]*A(x)^(-8)/4! +...
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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));polcoeff(A,n)}
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/* n-th Derivative: */
{Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D}
/* G.f.: [Paul D. Hanna, Dec 18 2010] */
{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,
A=exp(sum(m=0,n,Dx(m,x^(2*m+1)*A^(2*m+2))*A^(-2*m-2)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Jun 05 2025
A384574
G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^4).
Original entry on oeis.org
1, 1, 1, 5, 23, 155, 1236, 11286, 116333, 1329433, 16630343, 225606826, 3294976854, 51496560764, 856858516809, 15112857079891, 281479726839851, 5517842789917283, 113510479973132860, 2444032094604379100, 54948814775692303024, 1287258966133883349701
Offset: 0
-
a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-4*j+k, j)/(4*n-4*j+k)*a(n-j, j)));
A384575
G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^5).
Original entry on oeis.org
1, 1, 1, 6, 31, 236, 2166, 22722, 269889, 3567412, 51765431, 816476196, 13892821878, 253442895075, 4930644856063, 101830536332051, 2223767436058566, 51172807259226084, 1237092039069090235, 31332521053777095784, 829389782837272248191, 22894754438382163120136
Offset: 0
-
a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n-5*j+k, j)/(5*n-5*j+k)*a(n-j, j)));
A384576
G.f. A(x) satisfies A(x) = ( 1 + x * A(x*A(x))^(1/2) )^2.
Original entry on oeis.org
1, 2, 3, 8, 27, 118, 609, 3602, 23866, 174186, 1383868, 11860702, 108889022, 1064691402, 11034753421, 120739899232, 1389891203976, 16781698952902, 211959646629376, 2793804347189762, 38347179124969391, 547046497259184494, 8096627908313404104
Offset: 0
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a(n, k=2) = 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, j)));
Showing 1-6 of 6 results.