A213405 -id:A213405 - cp's OEIS frontend

A213403 G.f.: exp( Sum_{n>=1} binomial(6n-1, 3n) * x^n/n ).

1, 10, 281, 10580, 457700, 21475122, 1062749598, 54611328552, 2886091165052, 155866877884424, 8564415357567017, 477261537757290340, 26908911750685828972, 1532232857543951354044, 87987735421932575184876, 5089715542281323916803664, 296304273741441480224927436

Offset: 0

Paul D. Hanna, Jun 10 2012

nonn

			G.f.: A(x) = 1 + 10*x + 281*x^2 + 10580*x^3 + 457700*x^4 + 21475122*x^5 +...
such that A(x^3) = C(x)*C(u*x)*C(u^2*x) where u = exp(2*Pi*I/3) and
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500
Feihu Liu and Guoce Xin, Simple Generating Functions for Certain Young Tableaux with Periodic Walls, arXiv:2401.14627 [math.CO], 2024.

Cf. A000108, A079489, A213404, A213405, A213406.

{a(n)=polcoeff(exp(sum(m=1,n,binomial(6*m-1,3*m)*x^m/m)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

G.f. A(x) satisfies: A(x^3) = C(x)*C(u*x)*C(u^2*x) where u = exp(2*Pi*I/3) and C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).
Recurrence: (n-1)*n*(n+1)*(3*n-2)*(3*n-1)*(3*n+1)*(3*n+2)*(108*n^3 - 738*n^2 + 1626*n - 1145)*a(n) = 16*(n-1)*n*(3*n-2)*(3*n-1)*(15552*n^6 - 145152*n^5 + 544104*n^4 - 1066212*n^3 + 1170804*n^2 - 673396*n + 144935)*a(n-1) - 192*(n-1)*(1119744*n^9 - 16609536*n^8 + 108801792*n^7 - 413667648*n^6 + 1005574176*n^5 - 1616657184*n^4 + 1710678468*n^3 - 1140217942*n^2 + 429402110*n - 68310725)*a(n-2) + 2048*(2*n-5)*(2239488*n^9 - 39937536*n^8 + 314181504*n^7 - 1428637824*n^6 + 4127176800*n^5 - 7823974464*n^4 + 9674759436*n^3 - 7456006106*n^2 + 3201337522*n - 567810495)*a(n-3) - 1048576*(n-3)*(2*n-7)*(2*n-5)*(3*n-10)*(3*n-8)*(6*n-19)*(6*n-17)*(108*n^3 - 414*n^2 + 474*n - 149)*a(n-4). - Vaclav Kotesovec, Jul 05 2014
a(n) ~ (2-sqrt(2)*3^(1/4))*(1+sqrt(3)) * 2^(6*n+1) / (n^(3/2)*sqrt(3*Pi)). - Vaclav Kotesovec, Jul 05 2014

A213404 G.f.: exp( Sum_{n>=1} binomial(8n-1, 4n) * x^n/n ).

Original entry on oeis.org

1, 35, 3830, 570451, 98118690, 18345127262, 3621992085708, 743083237338755, 156855468465746346, 33846364485841559594, 7432235142547456907188, 1655432795976620159935790, 373110570133205997324473492, 84936332285861009708851200092, 19500719075082334054293510927128

Offset: 0

Paul D. Hanna, Jun 10 2012

nonn

			G.f.: A(x) = 1 + 35*x + 3830*x^2 + 570451*x^3 + 98118690*x^4 +...
such that A(x^4) = C(x)*C(-x)*C(I*x)*C(-I*x) where I^2 = -1 and
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...
Also, A(x^2) = G(x)*G(-x) where G(x) is the g.f. of A079489:
G(x) = 1 + 3*x + 22*x^2 + 211*x^3 + 2306*x^4 + 27230*x^5 + 338444*x^6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..410
Feihu Liu and Guoce Xin, Simple Generating Functions for Certain Young Tableaux with Periodic Walls, arXiv:2401.14627 [math.CO], 2024.

Cf. A000108, A079489, A213403, A213405, A213406.

{a(n)=polcoeff(exp(sum(m=1,n,binomial(8*m-1,4*m)*x^m/m)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

G.f. A(x) satisfies: A(x^4) = C(x)*C(-x)*C(I*x)*C(-I*x) where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

a(n) ~ (1-sqrt(2*(sqrt(2)-1))) * 4^(4*n+1) / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jul 05 2014

A213406 G.f.: exp( Sum_{n>=1} binomial(12n-1, 6n) * x^n/n ).

Original entry on oeis.org

1, 462, 782761, 1841287756, 5032296741620, 14989560797138774, 47213445715209298574, 154652100584276167220568, 521484200609508028036469644, 1798155951370712836530932544856, 6311247529513572335576033066558569, 22473253520120296968203645006140445948

Offset: 0

Paul D. Hanna, Jun 10 2012

nonn

			G.f.: A(x) = 1 + 462*x + 782761*x^2 + 1841287756*x^3 + 5032296741620*x^4 +...
such that A(x^6) = C(x)*C(u*x)*C(u^2*x)*C(-x)*C(-u*x)*C(-u^2*x) where u = exp(2*Pi*I/6) and
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...
Also, A(x^6) = G(x^3)*G(-x^3) where G(x) is the g.f. of A213403:
G(x) = 1 + 10*x + 281*x^2 + 10580*x^3 + 457700*x^4 + 21475122*x^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..275
Feihu Liu and Guoce Xin, Simple Generating Functions for Certain Young Tableaux with Periodic Walls, arXiv:2401.14627 [math.CO], 2024.

Cf. A000108, A079489, A213403, A213404, A213405.

{a(n)=polcoeff(exp(sum(m=1,n,binomial(12*m-1,6*m)*x^m/m)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

G.f. A(x) satisfies: A(x^6) = C(x)*C(u*x)*C(u^2*x)*C(u^3*x)*C(u^4*x)*C(u^5*x) where u = exp(2*Pi*I/6) and C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

a(n) ~ (2-sqrt(2)) * (2-sqrt(2)*3^(1/4)) * (sqrt(3)-1) * 8^(4*n+1) / (n^(3/2)*sqrt(3*Pi)). - Vaclav Kotesovec, Jul 05 2014

cp's OEIS Frontend

A213403 G.f.: exp( Sum_{n>=1} binomial(6n-1, 3n) * x^n/n ).

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A213404 G.f.: exp( Sum_{n>=1} binomial(8n-1, 4n) * x^n/n ).

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A213406 G.f.: exp( Sum_{n>=1} binomial(12n-1, 6n) * x^n/n ).

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cp's OEIS Frontend

A213403 G.f.: exp( Sum_{n>=1} binomial(6*n-1, 3*n) * x^n/n ).

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A213404 G.f.: exp( Sum_{n>=1} binomial(8*n-1, 4*n) * x^n/n ).

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A213406 G.f.: exp( Sum_{n>=1} binomial(12*n-1, 6*n) * x^n/n ).

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A213403 G.f.: exp( Sum_{n>=1} binomial(6n-1, 3n) * x^n/n ).

A213404 G.f.: exp( Sum_{n>=1} binomial(8n-1, 4n) * x^n/n ).

A213406 G.f.: exp( Sum_{n>=1} binomial(12n-1, 6n) * x^n/n ).