A349290 -id:A349290 - cp's OEIS frontend

A199475 G.f. satisfies A(x) = Sum_{n>=0} x^n * (1 - A(x)^(2*n+2))/(1 - A(x)^2).

1, 2, 7, 34, 195, 1225, 8146, 56336, 401005, 2918308, 21614216, 162385693, 1234515922, 9479336440, 73410868630, 572719097908, 4496923141241, 35509806367132, 281814387290431, 2246608404455588, 17982234787607464, 144458551104066553, 1164342291135424494

Offset: 0

Paul D. Hanna, Nov 08 2011

nonn

Compare to g.f. B(x) of A007317 (binomial transform of Catalan numbers):

B(x) = Sum_{n>=0} x^n * (1 - B(x)^(n+1))/(1 - B(x)).

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 34*x^3 + 195*x^4 + 1225*x^5 +...
where g.f. A = A(x) satisfies the equivalent expressions:
A = 1 + x*(1-A^4)/(1-A^2) + x^2*(1-A^6)/(1-A^2) + x^3*(1-A^8)/(1-A^2) +...
A = 1 + x*(1 + A^2) + x^2*(1 + A^2 + A^4) + x^3*(1 + A^2 + A^4 + A^6) +...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A007317, A171199, A199548, A349289, A349290, A349291, A349292, A349293.

Rest[CoefficientList[InverseSeries[Series[(2*x^2)/(1 + x^2 - Sqrt[1 - 4*x - 2*x^2 + x^4]), {x, 0, 30}], x], x]] (* Vaclav Kotesovec, Jul 30 2021 *)

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, A^(2*k))+x*O(x^n))); polcoeff(A, n)}

{a(n)=local(A=1+x);for(i=1,n,A=1/((1-x)*(1 - x*A^2+x*O(x^n))));polcoeff(A,n)}

{a(n)=polcoeff(1/x*serreverse(2*x^2/(1+x^2-sqrt(1-4*x-2*x^2+x^4+x^3*O(x^n)))),n)}

G.f. satisfies: A(x) = 1/((1-x)*(1 - x*A(x)^2)).
G.f.: A(x) = (1/x)*Series_Reversion( 2*x^2/(1+x^2 - sqrt(1-4*x-2*x^2+x^4)) ).
G.f. satisfies: A(x) = G(x*A(x)) and G(x) = A(x/G(x)) = g.f. of A171199, where G(x) = exp( Sum_{n>=1} [G(x)^n + G(x)^-n]*x^n/n ).
a(n) = 1 + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Jul 25 2021
a(n) ~ sqrt(387 + 35*sqrt(129)) * (35 + 3*sqrt(129))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Jul 30 2021
a(n) = Sum_{k=0..n} binomial(n+k,n-k) * binomial(3*k,k)/(2*k+1). - Seiichi Manyama, Oct 03 2023
D-finite with recurrence 2*n*(2*n+1)*a(n) +3*(-13*n^2+11*n-2)*a(n-1) +(35*n^2-23*n-42)*a(n-2) +(35*n^2-257*n+426)*a(n-3) +3*(-13*n^2+93*n-166)*a(n-4) +2*(n-4)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Feb 10 2024

A349289 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^3)).

Original entry on oeis.org

1, 2, 9, 61, 493, 4371, 41065, 401563, 4044097, 41658044, 436862457, 4648331765, 50057856881, 544557984498, 5975422922413, 66059269445451, 735064865871889, 8226310738656892, 92531697191189777, 1045551973586825023, 11862334695799444993

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002293, A007317, A199475, A346646, A349290, A349291, A349292, A349293.

A349289 := proc(n)
    add( binomial(n+2*k,3*k)*binomial(4*k,k)/(3*k+1),k=0..n)  ;
end proc:
seq(A349289(n),n=0..50) ; # R. J. Mathar, Feb 10 2024

nmax = 20; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^3)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 2 k, 3 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} binomial(n+2*k,3*k) * binomial(4*k,k) / (3*k+1).
a(n) = F([1/4, 1/2, 3/4, (1+n)/2, (2+n)/2, -n], [1/3, 2/3, 2/3, 1, 4/3], -2^10/3^6) where F is the generalized hypergeometric function. - Stefano Spezia, Nov 13 2021
a(n) ~ sqrt(1 + 2*r) / (4 * 2^(1/6) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/3)), where r = 0.0816785448577670972635343365300887975661075663022821172271... is the root of the equation 4^4 * r = 3^3 * (1-r)^3. - Vaclav Kotesovec, Nov 14 2021
D-finite with recurrence 81*n*(3*n-1)*(3*n+1)*a(n) +3*(243*n^3-7101*n^2+9986*n-3560)*a(n-1) +(-115027*n^3+514908*n^2-699869*n+269580)*a(n-2) +(-85543*n^3+1604715*n^2-6291692*n+6995280)*a(n-3) +(580211*n^3-6643158*n^2+23063299*n-23830944)*a(n-4) +(-33473*n^3-2231073*n^2+26352470*n-70945392)*a(n-5) +(-872129*n^3+17812344*n^2-119542699*n+264170868)*a(n-6) +(667171*n^3-14196243*n^2+100393472*n-236010000)*a(n-7) -6*(3*n-23)*(9948*n^2-147805*n+548868)*a(n-8) +4044*(3*n-26)*(n-8)*(3*n-22)*a(n-9)=0. - R. J. Mathar, Feb 10 2024
a(n) = 1 + Sum_{i, j, k, l>=0 and i+j+k+l=n-1} a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 10 2025

A349291 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^5)).

Original entry on oeis.org

1, 2, 13, 139, 1775, 24886, 370099, 5733304, 91518691, 1494815215, 24862931821, 419674102147, 7170713484877, 123783319369420, 2155542171446485, 37820343323942566, 667957770644685811, 11865421405897931581, 211856917750711562695, 3800040255017879663415

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002295, A007317, A199475, A346648, A349289, A349290, A349292, A349293.

nmax = 19; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^5)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 4 k, 5 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} binomial(n+4*k,5*k) * binomial(6*k,k) / (5*k+1).
a(n) ~ sqrt(1 + 4*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi*(1-r)) * n^(3/2) * r^(n + 1/5)), where r = 0.051436794119208432185504972091697516647... is the real root of the equation 6^6 * r = 5^5 * (1-r)^5. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 10 2025

A349292 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^6)).

Original entry on oeis.org

1, 2, 15, 190, 2871, 47643, 838888, 15389452, 290951545, 5629024955, 110908062511, 2217739684483, 44891645810124, 918086053852234, 18941156419798530, 393742848618632760, 8239112912485293357, 173406208518520952066, 3668419587671991125142

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002296, A007317, A199475, A346649, A349289, A349290, A349291, A349293.

nmax = 18; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^6)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 5 k, 6 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(7*k,k) / (6*k+1).
a(n) ~ sqrt(1 + 5*r) / (2 * 7^(2/3) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/6)), where r = 0.043408935906208378827553096713877784793679356... is the root of the equation 7^7 * r = 6^6 * (1-r)^6. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 11 2025

A349293 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^7)).

Original entry on oeis.org

1, 2, 17, 249, 4345, 83285, 1694273, 35915349, 784691569, 17545398747, 399545961817, 9234298584921, 216053290499201, 5107287712887563, 121795876378121121, 2926604574330886897, 70788399943851406825, 1722188546498276868124, 42114624858397590035177

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

nonn

In general, for k>=1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 + (k-1)*r) / ((k+1)^(1/k) * sqrt(2*k*(k+1)*Pi*(1-r)) * n^(3/2) * r^(n + 1/k)), where r is the smallest real root of the equation (k+1)^(k+1) * r = k^k * (1-r)^k. - Vaclav Kotesovec, Nov 14 2021

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A007317, A007556, A199475, A346650, A349289, A349290, A349291, A349292.

nmax = 18; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^7)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 6 k, 7 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]

a(n) = sum(k=0, n, binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1)); \\ Michel Marcus, Nov 14 2021

a(n) = Sum_{k=0..n} binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1).
a(n) ~ sqrt(1 + 6*r) / (2^(17/7) * sqrt(7*Pi*(1-r)) * n^(3/2) * r^(n + 1/7)), where r = 0.0375502499742240443056934699070050852345109331376051496159609551... is the real root of the equation 8^8 * r = 7^7 * (1-r)^7. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 11 2025

A349300 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^4)).

Original entry on oeis.org

1, 0, 1, 4, 21, 114, 651, 3844, 23301, 144169, 906866, 5782350, 37289431, 242793439, 1593918916, 10538988984, 70121101825, 469133993094, 3154115695476, 21299373321344, 144402246424591, 982506791975780, 6706724412165956, 45917245477282994

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002294, A005043, A346627, A346665, A349290, A349299, A349301, A349302, A349303.

nmax = 23; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[x]^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n + 3 k, 4 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 23}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1)); \\ Michel Marcus, Nov 14 2021

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1).
a(n) = (-1)^5*F([1/5, 2/5, 3/5, 4/5, (1+n)/3, (2+n)/3, (3+n)/3, -n], [1/4, 1/2, 1/2, 3/4, 3/4, 1, 5/4], 3^3*5^5/2^16), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 - 3*r) / (2 * 5^(3/4) * sqrt(2*Pi*(1+r)) * n^(3/2) * r^(n + 1/4)), where r = 0.136824361675510443450981569282313811786270109272790613523286... is the root of the equation 5^5 * r = 4^4 * (1+r)^4. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^4), where F(x) = Sum_{n >= 0} A002294(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^5. (End)

A366356 G.f. satisfies A(x) = 1/(1 - x) + x/A(x).

Original entry on oeis.org

1, 2, -1, 6, -17, 71, -292, 1284, -5807, 26961, -127627, 613815, -2990680, 14730714, -73229290, 366936232, -1851352819, 9397497759, -47957377933, 245903408245, -1266266092111, 6545667052321, -33954266444497, 176689391245147, -922112642288148, 4825154135801698

Offset: 0

Seiichi Manyama, Oct 08 2023

sign

Paolo Xausa, Table of n, a(n) for n = 0..1000

Partial sums give A363816.
Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366357, A366358, A366359.
Cf. A112478, A366363.

A366356[n_]:=(-1)^(n-1)Sum[Binomial[2k-1,k]Binomial[2k-1,n-k]/(2k-1),{k,0,n}];
Array[A366356,30,0] (* Paolo Xausa, Oct 20 2023 *)

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

G.f.: A(x) = -2*x*(1-x) / (1-sqrt(1+4*x*(1-x)^2)).

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

A366357 G.f. satisfies A(x) = 1/(1 - x) + x/A(x)^2.

Original entry on oeis.org

1, 2, -3, 19, -105, 690, -4781, 34708, -260189, 1999169, -15660175, 124596499, -1004110947, 8179379808, -67239070867, 557098881920, -4647368670949, 39001655222788, -329048378867467, 2789241880512899, -23743798316713367, 202894843070927860

Offset: 0

Seiichi Manyama, Oct 08 2023

sign

Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366356, A366358, A366359.

Cf. A364393, A366326, A366364.

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

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(3*k-1,k) * binomial(3*k-1,n-k)/(3*k-1).

A366358 G.f. satisfies A(x) = 1/(1 - x) + x/A(x)^3.

Original entry on oeis.org

1, 2, -5, 40, -319, 2908, -28151, 284908, -2977115, 31875709, -347884084, 3855802690, -43283239649, 491083601339, -5622489637406, 64877058557080, -753705528179423, 8808460811302729, -103487549564845199, 1221565052783161764, -14480208437556590345

Offset: 0

Seiichi Manyama, Oct 08 2023

sign

Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366356, A366357, A366359.

Cf. A364407, A366327, A366365.

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

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

A366359 G.f. satisfies A(x) = 1/(1 - x) + x/A(x)^4.

Original entry on oeis.org

1, 2, -7, 69, -715, 8351, -103735, 1346247, -18035023, 247520970, -3462344959, 49181268701, -707502644111, 10286493363184, -150913708053635, 2231345941617611, -33215679733509159, 497392118745778015, -7487512016559918595, 113242852989349372915

Offset: 0

Seiichi Manyama, Oct 08 2023

sign

Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366356, A366357, A366358.

Cf. A364408, A366328, A366366.

a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k-1, k)*binomial(5*k-1, n-k)/(5*k-1));

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k-1,k) * binomial(5*k-1,n-k)/(5*k-1).

cp's OEIS Frontend

A199475 G.f. satisfies A(x) = Sum_{n>=0} x^n * (1 - A(x)^(2*n+2))/(1 - A(x)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A349289 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^3)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A349291 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^5)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A349292 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^6)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A349293 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^7)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349300 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^4)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366356 G.f. satisfies A(x) = 1/(1 - x) + x/A(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366357 G.f. satisfies A(x) = 1/(1 - x) + x/A(x)^2.

Views

Author

Keywords

Crossrefs

Programs