A002293 -id:A002293 - cp's OEIS frontend

A380514 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

1, 1, 5, 67, 1537, 50021, 2107021, 108885295, 6665443457, 471522589417, 37843890892021, 3397250515809371, 337267132243022785, 36687625652474612557, 4339368321317331858557, 554467482301151809302151, 76112537023512618262963201, 11170667360636927554290623825, 1745500813880455301486766050917

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A002293, A069271, A380513, A380515, A380516.

a(n) = if(n==0, 1, 2*n!*sum(k=0, n-1, binomial(2*n+2*k, k)/((2*n+2*k)*(n-k-1)!)));

a(n) = 2 * n! * Sum_{k=0..n-1} binomial(2*n+2*k,k)/((2*n+2*k) * (n-k-1)!) for n > 0.

A382034 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^4), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 181, 5713, 246881, 13570081, 906180997, 71250724833, 6448375469665, 660286026034561, 75472025139452261, 9525947428687403473, 1315935073971181422721, 197485196722573989608289, 31993978774204625549549221, 5565216938342017912128576961, 1034506012356981473110554574145

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A161630, A382032, A382033.

Cf. A002293, A377630.

a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (k+1)^(n-k-1)*binomial(4*n, k)/(n-k-1)!));

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(4*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377630. F(x) = log(A(x))/x = B(x*A(x))^4.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^4 ) ).

A153396 G.f.: A(x) = F(xG(x)^3) where F(x) = G(x/F(x)^2) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)^2) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 5, 32, 228, 1726, 13587, 109923, 907499, 7609898, 64609346, 554108863, 4792190298, 41739160686, 365746143064, 3221723465187, 28509044813580, 253295607463902, 2258539046009268, 20203103111671575, 181242298665210280

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

			G.f.: A(x) = F(x*G(x)^3) = 1 + x + 5*x^2 + 32*x^3 + 228*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +...
G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 74*x^3 + 545*x^4 + 4228*x^5 +...
G(x)^3*A(x)^2 = 1 + 5*x + 32*x^2 + 228*x^3 + 1726*x^4 + 13587*x^5 +...

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A000108, A001764, A002293; A153395, A153397, A093341.

Join[{1},Table[Sum[Binomial[2k+1,k]/(2k+1) Binomial[4n-k,n-k]3 k/(4n-k), {k,0,n}],{n,20}]] (* Harvey P. Dale, Feb 09 2015 *)

{a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+1,k)/(2*k+1)*binomial(4*(n-k)+3*k,n-k)*3*k/(4*(n-k)+3*k)))}

a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(4n-k,n-k)*3k/(4n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^2 where G(x) is the g.f. of A002293.
G.f. satisfies: A(x/F(x)^2) = F(x*F(x)) where F(x) is the g.f. of A000108.
G.f. satisfies: A(x/H(x)) = F(x*H(x)^2) where H(x) = 1 + x*H(x)^3 is the g.f. of A001764 and F(x) is the g.f. of A000108.
G.f. satisfies: A(-x*A(x)^9) = 1/A(x). - Alexander Burstein, Apr 14 2020
Recurrence: 243*(n-1)*n*(n+1)*(3*n - 5)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(147456*n^6 - 1998336*n^5 + 11209920*n^4 - 33294250*n^3 + 55173779*n^2 - 48321229*n + 17452260)*a(n) = 72*(n-1)*n*(3*n - 5)*(3*n - 4)*(127401984*n^9 - 2045067264*n^8 + 14240360448*n^7 - 56278911936*n^6 + 138595592064*n^5 - 219567715966*n^4 + 222542820712*n^3 - 138190518059*n^2 + 47259501167*n - 6683489400)*a(n-1) - 48*(n-1)*(16307453952*n^12 - 351459606528*n^11 + 3428587929600*n^10 - 20001961205760*n^9 + 77643945578496*n^8 - 211031837008384*n^7 + 411217026027200*n^6 - 577827896836090*n^5 + 579810023200127*n^4 - 403994885007838*n^3 + 184802213339825*n^2 - 49548085570200*n + 5838168798000)*a(n-2) + 128*(2*n - 5)*(4*n - 11)*(4*n - 9)*(8*n - 23)*(8*n - 21)*(8*n - 19)*(8*n - 17)*(147456*n^6 - 1113600*n^5 + 3430080*n^4 - 5488810*n^3 + 4779029*n^2 - 2123685*n + 369600)*a(n-3). - Vaclav Kotesovec, Feb 22 2015
a(n) ~ (256/27)^n / n^(5/4) * (3^(1/4)*sqrt(EllipticK(1/sqrt(2)))/(2*Pi)^(3/4) - sqrt(3/(2*Pi))/n^(1/4) + (2/(3*Pi))^(1/4) / sqrt(EllipticK(1/sqrt(2)))/n^(1/2)), where EllipticK(1/sqrt(2)) = A093341 = GAMMA(1/4)^2/(4*(Pi)^(1/2)) = 1.85407467730137191843385... (= EllipticK[1/2] in Mathematica). - Vaclav Kotesovec, Feb 22 2015

A381916 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / B(x) ), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 4, 29, 270, 2897, 34051, 426199, 5582619, 75660075, 1052748518, 14956346820, 216088986290, 3165555750458, 46912569559556, 702072705679590, 10595488626535181, 161071258091631337, 2464201011094137000, 37911236702465987337, 586166246311185676045, 9103432675706477369934

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381914, A381915.
Cf. A381880, A381913.
Cf. A002293.

a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(4*n-k+2, n-k)/(n+4*k+1));

G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x*A(x))^3.

a(n) = Sum_{k=0..n} binomial(n+4*k+1,k) * binomial(4*n-k+2,n-k)/(n+4*k+1).

A382031 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x)^2)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 43, 1177, 46681, 2419291, 154587427, 11735209585, 1031418915121, 102979800567091, 11510663862332251, 1423811747933017609, 193073662118499898633, 28479005472094048953355, 4539456019668776334683731, 777538096585429376795405281, 142419954152382631361835929185

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A212722, A382029, A382030.
Cf. A380513, A382016.
Cf. A002293, A382044.

a(n) = if(n==0, 1, n!*sum(k=0, n-1, (2*k+1)^(n-k-1)*binomial(n+3*k, k)/((n+3*k)*(n-k-1)!)));

Let F(x) be the e.g.f. of A382044. F(x) = log(A(x))/x = B(x*A(x)^2).

a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+3*k,k)/((n+3*k) * (n-k-1)!) for n > 0.

A382038 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^4) ), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 11, 244, 8285, 381096, 22175167, 1562582848, 129381990201, 12313784396800, 1324663415429651, 158957183013686784, 21051725357219126869, 3050121640032545419264, 479928476696367747954375, 81499293517054315684642816, 14856515462975583258374526833, 2893604521320117995839047401472

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A052873, A382036, A382037.

Cf. A002293, A382034.

a(n) = if(n==0, 1, (n-1)!*sum(k=0, n-1, (n+1)^(n-k-1)*binomial(4*n, k)/(n-k-1)!));

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * B(x*A(x))^4).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(4*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^4 ) ).

A137572 The first upper diagonal of square array A137570; equals the convolution of the main diagonal A137571 with A002293.

Original entry on oeis.org

1, 3, 16, 100, 681, 4908, 36842, 285158, 2260257, 18257902, 149769225, 1244277499, 10448404901, 88538107802, 756153001241, 6501989278168, 56244305146039, 489111092027854, 4273491476147117, 37496699100314116, 330261353255659842

Offset: 0

Paul D. Hanna, Jan 27 2008

nonn

			G.f.: A(x) = 1 + 3*x + 16*x^2 + 100*x^3 + 681*x^4 + 4908*x^5 +...;
A(x) = F(x)/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].

Cf. A137570, A137571, A137573; A000108, A002293.

{a(n)=local(m=n+1,C,F,A); C=Ser(vector(m,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))); A=F/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}

G.f. A(x) = F(x)/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108) and F(x) = 1 + xF(x)^4 is g.f. of A002293.

A137573 The first lower diagonal in square array A137570; equals the convolution of the main diagonal A137571 with the Catalan numbers (A000108) and with the square of A002293.

Original entry on oeis.org

1, 5, 29, 186, 1281, 9294, 70109, 544833, 4333381, 35108351, 288738813, 2404256945, 20228988678, 171716799066, 1468804301441, 12647321103329, 109538312419238, 953622158606749, 8340394595266367, 73247287493299642

Offset: 0

Paul D. Hanna, Jan 27 2008

nonn

			G.f.: A(x) = 1 + 5*x + 29*x^2 + 186*x^3 + 1281*x^4 + 9294*x^5 +...;
A(x) = C(x)*F(x)^2/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].

Cf. A137570, A137571, A137572; A000108, A002293.

{a(n)=local(m=n+1,C,F,A); C=Ser(vector(m,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))); A=C*F^2/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}

G.f. A(x) = C(x)*F(x)^2/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108) and F(x) = 1 + xF(x)^4 is g.f. of A002293.

A363111 Expansion of g.f. A(x) = F(xF(x)^7), where F(x) = 1 + xF(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 11, 127, 1547, 19652, 258069, 3481034, 47999915, 674086924, 9612919156, 138878011335, 2028718584989, 29918897595468, 444889269572286, 6663228661354420, 100430376524360459, 1522215623202615036, 23187346871707554564, 354783440893854307244

Offset: 0

Paul D. Hanna, May 30 2023

nonn

Compare the g.f. A(x) = F(x*F(x)^7) to F(-x*F(x)^7) = 1/F(x), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

			G.f.: A(x) = 1 + x + 11*x^2 + 127*x^3 + 1547*x^4 + 19652*x^5 + 258069*x^6 + 3481034*x^7 + 47999915*x^8 + 674086924*x^9 + ...
such that A(x) = F(x*F(x)^7) where F(x) = 1 + x*F(x)^4 begins
F(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 + 53820*x^7 + ... + A002293(n)*x^n + ...
RELATED SERIES.
Let B(x) = A(x*B(x)^3) = ( Series_Reversion( x/A(x)^3 )/x )^(1/3) which begins
B(x) = 1 + x + 14*x^2 + 238*x^3 + 4578*x^4 + 95130*x^5 + 2082150*x^6 + 47295990*x^7 + 1104598378*x^8 + ...
then
( (B(x) - 1)/x )^(1/7) = 1 + 2*x + 22*x^2 + 350*x^3 + 6538*x^4 + 133658*x^5 + 2895214*x^6 + 65294502*x^7 + ... + A363304(n)*x^n + ...
is an integer series.

Cf. A363304, A363308, A363309, A002293.

{a(n) = if(n==0, 1, sum(k=1, n, 7*k* binomial(4*k+1, k) * binomial(4*n+3*k, n-k) / ((4*k+1)*(4*n+3*k)) ) )}
for(n=0, 30, print1(a(n), ", "))

/* G.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 */
{a(n) = my(F = 1); for(i=1,n, F = 1 + x*F^4 + x*O(x^n));
polcoeff( subst(F, x, x*F^7), n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows; here, F(x) is the g.f. of A002293.
(1) A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4.
(2) A(x) = B(x/A(x)^3) where B(x) = A(x*B(x)^3) = F( x*B(x)^3 * F(x*B(x)^3)^7 ).
(3) a(n) = Sum_{k=1..n} 7*k* binomial(4*k+1, k) * binomial(4*n+3*k, n-k) / ((4*k+1)*(4*n+3*k)) for n > 0, with a(0) = 1.

A381914 Expansion of (1/x) * Series_Reversion( x * (1-x) / B(x) ), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 2, 10, 72, 624, 6009, 61809, 664813, 7384613, 84045565, 974913510, 11483316680, 136974177209, 1651166320547, 20083352214058, 246168280262403, 3037682020219285, 37706043912831337, 470482875049515074, 5897864081341146065, 74243055437832292562, 938101296155866961124

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A381915, A381916.
Cf. A381817, A381911.
Cf. A002293.

a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(2*n-k, n-k)/(n+4*k+1));

G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x*A(x)).

a(n) = Sum_{k=0..n} binomial(n+4*k+1,k) * binomial(2*n-k,n-k)/(n+4*k+1).

cp's OEIS Frontend

A380514 Expansion of e.g.f. exp(x*G(x)^2) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A382034 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^4), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A153396 G.f.: A(x) = F(x*G(x)^3) where F(x) = G(x/F(x)^2) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)^2) = 1 + x*G(x)^4 is the g.f. of A002293.

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A381916 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / B(x) ), where B(x) is the g.f. of A002293.

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A382031 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x)^2)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A382038 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^4) ), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A137572 The first upper diagonal of square array A137570; equals the convolution of the main diagonal A137571 with A002293.

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A137573 The first lower diagonal in square array A137570; equals the convolution of the main diagonal A137571 with the Catalan numbers (A000108) and with the square of A002293.

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A363111 Expansion of g.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

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A380514 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

A382034 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^4), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

A153396 G.f.: A(x) = F(xG(x)^3) where F(x) = G(x/F(x)^2) = 1 + xF(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(xG(x)^2) = 1 + xG(x)^4 is the g.f. of A002293.

A382031 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x)^2)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

A363111 Expansion of g.f. A(x) = F(xF(x)^7), where F(x) = 1 + xF(x)^4 is the g.f. of A002293.