A002295 -id:A002295 - cp's OEIS frontend

A346767 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(6k,k) / (5k + 1).

1, 1, 7, 70, 855, 11907, 182714, 3029040, 53565875, 1001599339, 19674910572, 404009742858, 8638256718929, 191702754433132, 4403979321915615, 104496256532120370, 2555972287817569101, 64340126437548435175, 1664318438781195696512, 44182488823505663971205

Offset: 0

Ilya Gutkovskiy, Aug 02 2021

nonn

Stirling transform of A002295.

Michael De Vlieger, Table of n, a(n) for n = 0..499

Cf. A002295, A064856, A346764, A346765, A346766, A346768, A346769.

Table[Sum[StirlingS2[n, k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 19; CoefficientList[Series[HypergeometricPFQ[{1/6, 1/3, 1/2, 2/3, 5/6}, {2/5, 3/5, 4/5, 1, 6/5}, 46656 (Exp[x] - 1)/3125], {x, 0, nmax}], x] Range[0, nmax]!

a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(6*k, k)/(5*k + 1)); \\ Michel Marcus, Aug 02 2021

G.f.: Sum_{k>=0} ( binomial(6*k,k) / (5*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).

A349301 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^5)).

Original entry on oeis.org

1, 0, 1, 5, 31, 200, 1351, 9430, 67531, 493505, 3665981, 27602081, 210179437, 1615820402, 12524590873, 97775503808, 768083233899, 6067097140799, 48159634951855, 383965003803985, 3073379977522321, 24688458872260007, 198968304164411309

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

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

Cf. A002295, A005043, A346627, A346666, A349291, A349299, A349300, A349302, A349303.

nmax = 22; 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[(-1)^(n - k) Binomial[n + 4 k, 5 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 22}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * 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 = 0.11589193448796656683553561932282471511616945026903125324... is the real root of the equation 6^6 * r = 5^5 * (1+r)^5. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^5), where F(x) = Sum_{n >= 0} A002295(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^6. (End)

A364866 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)^5).

Original entry on oeis.org

1, 1, 4, 21, 124, 781, 5120, 34474, 236492, 1644222, 11543644, 81623504, 580104672, 4137414963, 29574658416, 211639869236, 1514729242092, 10832683182538, 77342204972120, 550791674067623, 3908735530965604, 27612614422978557, 193943797650498016

Offset: 0

Seiichi Manyama, Aug 11 2023

sign

a(34) is negative.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A291534, A364864, A364865, A365218.

Cf. A002295.

a(n) = sum(k=0, n, (-1)^k*2^(n-k)*binomial(n, k)*binomial(5*n+k+1, n)/(5*n+k+1));

a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(5*n+k+1,n) / (5*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(5*n,k-1) for n > 0.

A365185 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)).

Original entry on oeis.org

1, 1, 6, 46, 411, 3996, 41062, 438662, 4823133, 54221518, 620404859, 7201317005, 84590041441, 1003656037278, 12010861830069, 144804336388912, 1757106190680819, 21443109365898743, 263009775111233392, 3240530659303505547, 40088688455992604594

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A365184, A365186, A365187, A365188, A365189.

Cf. A364748.

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

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

A365187 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 6, 48, 446, 4511, 48218, 535800, 6127598, 71648868, 852668952, 10293847592, 125759270354, 1551872951050, 19314892116764, 242182938963024, 3056337851481678, 38790948190319404, 494825459824571528, 6340628082364678016, 81577931200018721464

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A365184, A365185, A365186, A365188, A365189.

Cf. A365193.

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

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

A365188 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 6, 49, 465, 4807, 52533, 596936, 6981798, 83497115, 1016367737, 12550853210, 156845913315, 1979870172453, 25207383853375, 323325558146400, 4174108907656633, 54195445136831670, 707225283913589280, 9270735916525207605, 122020617365557674605

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A365184, A365185, A365186, A365187, A365189.

Cf. A243667.

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

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

A386368 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k-2,n-k-1).

Original entry on oeis.org

0, 1, 16, 246, 3736, 56421, 849432, 12763878, 191548464, 2871970110, 43031833656, 644432826478, 9646983339456, 144366433138955, 2159869510669320, 32306874783230556, 483151884326658144, 7224464127509984490, 108011596038055519680, 1614676987907480393940

Offset: 0

Seiichi Manyama, Jul 19 2025

nonn

			(1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ) = x + 8*x^2 + 82*x^3 + 934*x^4 + 56421*x^5/5 + ...

Cf. A000302, A036829, A308523, A386367.
Cf. A079679, A386567, A386615, A386616.
Cf. A002295, A130564.

A386368 := proc(n::integer)
    add(binomial(6*k,k)*binomial(6*n-6*k-2,n-k-1),k=0..n-1) ;
end proc:
seq(A386368(n),n=0..80) ; # R. J. Mathar, Jul 30 2025

a(n) = sum(k=0, n-1, binomial(6*k, k)*binomial(6*n-6*k-2, n-k-1));

my(N=20, x='x+O('x^N), g=x*sum(k=0, N, binomial(6*k+4, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-6*g)^2))

G.f.: g*(1-g)/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ).
G.f.: (g-1)/(6-5*g)^2 where g=1+x*g^6.
a(n) = Sum_{k=0..n-1} binomial(6*k-2+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n-1,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k-1,k).
Conjecture D-finite with recurrence 48828125*(n-1)*(5*n-4)*(5*n-3) *(432862082629612805*n -769306661967834399) *(5*n-2)*(5*n-1)*a(n) +1125000*(-405245406115816219575000*n^6 +2613180799468910510392500*n^5 -7667164406968651479521250*n^4 +13834502135358262506660375*n^3 -16251583347734702117341345*n^2 +11251247074043948959380314*n -3395699069351241765495720)*a(n-1) +33592320*(142281690918326440537500*n^6 -1266424338521609272012500*n^5 +5236041263583271687953750*n^4 -12786608152035075786775875*n^3 +18838556229131595646260055*n^2 -15323925851720394901667853*n +5240681406952416812161236)*a(n-2) -53444359913472*(6*n-17) *(395547729523405*n -538181211711288)*(6*n-13) *(3*n-7)*(2*n-5) *(3*n-8)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A110448 G.f.: A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), where A056045(n) = Sum_{d|n} binomial(n,d).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 18, 23, 49, 73, 145, 194, 474, 611, 1331, 2027, 4393, 5919, 14736, 19415, 46487, 68504, 156618, 212055, 560380, 739165, 1833012, 2657837, 6513367, 8743208, 23649777, 31140300, 81276046, 114962333, 293600318, 391926154

Offset: 0

Paul D. Hanna, Jul 20 2005, Nov 10 2007

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 18*x^6 +...
where A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), or
A(x) = exp(x + 3/2*x^2 + 4/3*x^3 + 11/4*x^4 + 6/5*x^5 +...).
The g.f. can also be expressed as the product:
A(x) = 1/(1-x)*G000108(x^2)*G001764(x^3)*G002293(x^4)*G002294(x^5)*...
where the functions are g.f.s of well-known sequences:
G000108(x) = 1 + x*G000108(x)^2 = g.f. of A000108 ;
G001764(x) = 1 + x*G001764(x)^3 = g.f. of A001764 ;
G002293(x) = 1 + x*G002293(x)^4 = g.f. of A002293 ;
G002294(x) = 1 + x*G002294(x)^5 = g.f. of A002294 ; etc.

Seiichi Manyama, Table of n, a(n) for n = 0..3337

Cf. A056045, A174461, A206290.

Cf. A000108, A001764, A002293, A002294, A002295.

{a(n)=polcoeff(exp(x*Ser(vector(n,m, sumdiv(m,d,binomial(m,d))/m))+x*O(x^n)),n)}

{a(n)=polcoeff(prod(m=1,n,1/x*serreverse(x/(1+x^m +x*O(x^n)))),n)}

G.f.: A(x) = Product_{n>=1} (1/x)*Series_Reversion( x/(1 + x^n) ); equivalently, G.f.: A(x) = Product_{n>=1} G(x^n,n) where G(x,n) = 1 + x*G(x,n)^n.

a(n) ~ c * 2^n / n^(3/2), where c = 2.8176325363130737043447... if n is even and c = 1.784372019603712867208... if n is odd. - Vaclav Kotesovec, Jan 15 2019

A346580 a(n) = (1/(6n)) Sum_{d|n} mu(n/d) * binomial(6*d,d).

Original entry on oeis.org

1, 5, 45, 440, 4750, 54081, 642341, 7861216, 98480286, 1256564750, 16273981757, 213378921432, 2826867619108, 37782552518473, 508840821825750, 6898459208449920, 94070535317459017, 1289430373107917718, 17755914760643605781, 245518560759177014000, 3407586451859019939012

Offset: 1

Ilya Gutkovskiy, Jul 24 2021

nonn

Inverse Euler transform of A002295.

Moebius transform of A261499.

Cf. A002295, A004355, A008683, A022553, A261499, A346577, A346578, A346579, A346581, A346582.

Table[(1/(6 n)) Sum[MoebiusMu[n/d] Binomial[6 d, d], {d, Divisors[n]}], {n, 21}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(6*d,d))/(6*n); \\ Michel Marcus, Jul 24 2021

A346682 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(6k,k) / (5k + 1).

Original entry on oeis.org

1, 0, 6, 45, 461, 5020, 57812, 691586, 8512048, 107095262, 1371219004, 17808830924, 234048288772, 3106795261083, 41593689788637, 560980967638479, 7614970691479315, 103957059568762775, 1426355910771621805, 19658792867492660060, 272046427837226505466

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..856

Cf. A002295, A032357, A188678, A346065, A346666.

Cf. A346680, A346681, A346683, A346684.

Table[Sum[(-1)^(n - k) Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^5 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(6*k, k)/(5*k + 1)); \\ Michel Marcus, Jul 29 2021

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^5 * A(x)^6.

a(n) ~ 2^(6*n + 6) * 3^(6*n + 13/2) / (49781 * sqrt(Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

cp's OEIS Frontend

A346767 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(6*k,k) / (5*k + 1).

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A349301 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^5)).

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A364866 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 + x*A(x)^5).

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A365185 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)).

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A365187 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)^3).

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A365188 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)^4).

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A386368 a(n) = Sum_{k=0..n-1} binomial(6*k,k) * binomial(6*n-6*k-2,n-k-1).

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A110448 G.f.: A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), where A056045(n) = Sum_{d|n} binomial(n,d).

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A346580 a(n) = (1/(6*n)) * Sum_{d|n} mu(n/d) * binomial(6*d,d).

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A346767 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(6k,k) / (5k + 1).

A364866 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)^5).

A365185 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)).

A365187 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^3).

A365188 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^4).

A386368 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k-2,n-k-1).

A346580 a(n) = (1/(6n)) Sum_{d|n} mu(n/d) * binomial(6*d,d).

A346682 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(6k,k) / (5k + 1).