A396097 - OEIS

%I #14 May 29 2026 15:06:06

%S 1,3,24,287,4082,63854,1060893,18375481,328140158,5998110712,

%T 111679078694,2110614752092,40383196151858,780713537950989,

%U 15227082192168750,299259411611190032,5920512681769082585,117815581575201608792,2356618411595244754811,47356204772093822680868,955566142938585617722546

%N G.f. A(x) satisfies Sum_{k=0..n} [x^k] A(x)^n = Sum_{k=0..2*n} [x^k] C(x)^n for n >= 0, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan function.

%C Compare to: Sum_{k=0..n} [x^k] 1/(1-x)^n = (2*n)!/n!^2 = A000984(n).

%C Compare to: Sum_{k=0..n} [x^k] 1/(1-x)^(2*n) = Sum_{k=0..2*n} [x^k] 1/(1-x)^n = (3*n)!/(n!*(2*n)!) = A005809(n).

%H Paul D. Hanna, <a href="/A396097/b396097.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) Sum_{k=0..n} [x^k] A(x)^n = Sum_{k=0..2*n} [x^k] C(x)^n for n >= 0, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan function.

%F (2) Sum_{k=0..n} [x^k] A(x)^n = A352275(n) where A352275(n) = Sum_{k=0..2*n} binomial(n + 2*k,k) * n/(n + 2*k) for n >= 0.

%F (3) A(x) = G(x/A(x)) where G(x) = A(x*G(x)) satisfies (G(x) + x*G'(x)) / (G(x) - x*G(x)^2) = Sum_{n>=0} A352275(n) * x^n.

%F (4) A(x) = (1-x) * (A(x) - x*A'(x)) * Sum_{n>=0} A352275(n) * x^n / A(x)^n.

%F From _Vaclav Kotesovec_, May 29 2026: (Start)

%F a(n) ~ c * d^n / n^(3/2), where

%F d = 21.7870041673292333577648845136818182882906239511526041767826595572589580609... and

%F c = 0.14785584766568055166299618504433412209227083010399493968555118456367388152...

%F Conjecture: d is the largest real root of the equation -132400 + 2959788*d - 23123461*d^2 + 55640000*d^3 - 43613528*d^4 + 6629032*d^5 - 490000*d^6 + 12500*d^7 == 0. (End)

%e G.f.: A(x) = 1 + 3*x + 24*x^2 + 287*x^3 + 4082*x^4 + 63854*x^5 + 1060893*x^6 + 18375481*x^7 + 328140158*x^8 + 5998110712*x^9 + 111679078694*x^10 + ...

%e RELATED SERIES.

%e The series G(x) = A(x*G(x)) begins

%e G(x) = 1 + 3*x + 33*x^2 + 530*x^3 + 10055*x^4 + 209357*x^5 + 4625679*x^6 + 106500731*x^7 + 2527389563*x^8 + ...

%e satisfies (G(x) + x*G'(x)) / (G(x) - x*G(x)^2) = D(x) where

%e D(x) = 1 + 4*x + 64*x^2 + 1429*x^3 + 35072*x^4 + 898129*x^5 + 23571781*x^6 + 628750217*x^7 + ... + A352275(n)*x^n + ...

%e Thus, G(x) = 1 + Integral G(x)*(D(x)-1)/x - D(x)*G(x)^2 dx

%e and A(x) = x/Series_Reversion(x*G(x)).

%e RELATED TABLES.

%e We shall compare the terms of the following tables to illustrate the defining property of the g.f. of this sequence.

%e Given the g.f. A(x), the table of coefficients of x^k in A(x)^n begins

%e   n = 1: [1,  3,  24,  287,  4082,   63854,  1060893, ...];

%e   n = 2: [1,  6,  57,  718, 10462,  165976,  2783215, ...];

%e   n = 3: [1,  9,  99, 1320, 19788,  319299,  5414988, ...];

%e   n = 4: [1, 12, 150, 2120, 32789,  539348,  9266334, ...];

%e   n = 5: [1, 15, 210, 3145, 50275,  844483, 14719195, ...];

%e   n = 6: [1, 18, 279, 4422, 73137, 1256142, 22237782, ...];

%e   ...

%e Given the Catalan series C(x) = 1 + x*C(x)^2, the table of coefficients of x^k in C(x)^n begins

%e   n = 1: [1, 1,  2,  5,  14,  42,  132,  429,  1430, ...];

%e   n = 2: [1, 2,  5, 14,  42, 132,  429, 1430,  4862, ...];

%e   n = 3: [1, 3,  9, 28,  90, 297, 1001, 3432, 11934, ...];

%e   n = 4: [1, 4, 14, 48, 165, 572, 2002, 7072, 25194, ...];

%e   ...

%e From these two tables, we see that the following partial sums of the rows in the above tables are equal

%e   n = 1: (1 + 3) = (1 + 1 + 2);

%e   n = 2: (1 + 6 + 57) = (1 + 2 + 5 + 14 + 42);

%e   n = 3: (1 + 9 + 99 + 1320) = (1 + 3 + 9 + 28 + 90 + 297 + 1001);

%e   n = 4: (1 + 12 + 150 + 2120 + 32789) = (1 + 4 + 14 + 48 + 165 + 572 + 2002 + 7072 + 25194);

%e   ...

%o (PARI) /* By Definition (slow): */

%o {d(n) = if(n==0,1, sum(k=0,2*n, binomial(n + 2*k,k) * n/(n + 2*k) ) )}

%o {a(n) = if(n==0, 1, (d(n) - sum(k=0, n, polcoef(sum(j=0, min(k, n-1), a(j)*x^j)^n + x*O(x^k), k)))/n)}

%o for(n=0,12,print1(a(n),", "))

%o (PARI) /* Faster, using series reversion: */

%o {d(n) = if(n==0,1, sum(k=0,2*n, binomial(n + 2*k,k) * n/(n + 2*k) ) )}

%o {a(n) = my(D = sum(k=0, n, d(k)*x^k) +x^3*O(x^n), G=1+x*O(x^n));

%o for(i=1, n, G = 1 + intformal( (D-1)*G/x - D*G^2));

%o polcoef(GF = x/serreverse(x*G +x^2*O(x^n)), n)}

%o {upto(n) = a(n); Vec(GF)}

%o upto(30)

%Y Cf. A352275, A232683, A000108.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 28 2026