A002294 -id:A002294 - cp's OEIS frontend

A386699 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n,k).

1, 7, 69, 733, 8061, 90462, 1028871, 11814376, 136643085, 1589311381, 18569375114, 217773347502, 2561944357311, 30219704365104, 357278540928168, 4232449819704768, 50227362114232109, 596988743410929087, 7105534815529752831, 84678089652554263155, 1010268312800732117946

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A000244, A100192, A385004, A385498.
Cf. A163455, A371739, A386702.
Cf. A002294.

Table[(243/16)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, -1/2]), {n,0,25}] (* Vaclav Kotesovec, Jul 30 2025 *)

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

a(n) = [x^n] 1/((1-3*x) * (1-x)^(4*n)).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 128*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(3052*n^4 - 15114*n^3 + 26432*n^2 - 18693*n + 4131)*a(n) = 8*(42807352*n^8 - 285737492*n^7 + 758983420*n^6 - 1002945218*n^5 + 644348866*n^4 - 111879380*n^3 - 84004497*n^2 + 44187381*n - 5806080)*a(n-1) - 1215*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(3052*n^4 - 2906*n^3 - 598*n^2 + 1037*n - 192)*a(n-2).
a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi*n) * 2^(8*n + 1/2)). (End)
G.f.: g/((3-2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^4*(15-8*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

A206290 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/(1 + x^k) ).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 17, 29, 44, 77, 114, 218, 330, 617, 987, 1913, 2968, 6068, 9500, 19263, 31399, 64268, 101702, 218891, 348559, 735823, 1205239, 2576727, 4119884, 9100854, 14588992, 31841260, 52163378, 114485092, 183947681, 414704366, 667453931, 1487920000

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Compare to the g.f. of partition numbers (A000041): Sum_{n>=0} Product_{k=1..n} x/(1 - x^k).

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 12*x^6 + 17*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n( x/(1 + x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + x^3 + x^4 + x^5 + x^6 +...+ x^(n+1) +...
G_2(x) = x + x^3 + 2*x^5 + 5*x^7 + 14*x^9 +...+ A000108(n)*x^(2*n+1) +...
G_3(x) = x + x^4 + 3*x^7 + 12*x^10 + 55*x^13 +...+ A001764(n)*x^(3*n+1) +...
G_4(x) = x + x^5 + 4*x^9 + 22*x^13 + 140*x^17 +...+ A002293(n)*x^(4*n+1) +...
G_5(x) = x + x^6 + 5*x^11 + 35*x^16 + 285*x^21 +...+ A002294(n)*x^(5*n+1) +...
G_6(x) = x + x^7 + 6*x^13 + 51*x^19 + 506*x^25 +...+ A002295(n)*x^(6*n+1) +...
G_7(x) = x + x^8 + 7*x^15 + 70*x^22 + 819*x^29 +...+ A002296(n)*x^(7*n+1) +...
Note that G_n(x) = x + x*G_n(x)^n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A206289, A194560, A110448.

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x/(1+x^k+x*O(x^n))))),n)}
for(n=0,45,print1(a(n),", "))

G.f.: Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:
(1) G_n(x) = Series_Reversion( x/(1 + x^n) ),
(2) G_n(x) = x + x*G_n(x)^n,
(3) G_n(x) = Sum_{k>=0} binomial(n*k+1, k) * x^(n*k+1) / (n*k+1).

A235534 a(n) = binomial(6n, 2n) / (4*n + 1).

Original entry on oeis.org

1, 3, 55, 1428, 43263, 1430715, 50067108, 1822766520, 68328754959, 2619631042665, 102240109897695, 4048514844039120, 162250238001816900, 6568517413771094628, 268225186597703313816, 11034966795189838872624, 456949965738717944767791

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=4, k=2 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.

First bisection of A001764.

Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), this sequence (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), A235535 (l=6, k=3).

l:=4; k:=2; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* where l is divisible by all the prime factors of k */

Table[Binomial[6 n, 2 n]/(4 n + 1), {n, 0, 20}]

a(n) = A047749(4*n-2) for n>0.
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 4F3(1/6,1/3,2/3,5/6; 1/2,3/4,5/4; 729*x/16).
a(n) ~ 3^(6*n+1/2)/(sqrt(Pi)*2^(4*n+7/2)*n^(3/2)). (End)

A235535 a(n) = binomial(9n, 3n) / (6*n + 1).

Original entry on oeis.org

1, 12, 1428, 246675, 50067108, 11124755664, 2619631042665, 642312451217745, 162250238001816900, 41932353590942745504, 11034966795189838872624, 2946924270225408943665279, 796607831560617902288322405, 217550867863011281855594752680

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=6, k=3 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.

Also, the sequence follows A002296 and A235536, namely binomial(7*n,n)/(6*n+1) and binomial(8*n,2*n)/(6*n+1); naturally, even binomial(10*n,4*n)/(6*n+1) is always integer.

Robert Israel, Table of n, a(n) for n = 0..403
Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), this sequence (l=6, k=3).

l:=6; k:=3; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* here l is divisible by all the prime factors of k */

seq(binomial(9*n,3*n)/(6*n+1), n=0..30); # Robert Israel, Feb 15 2021

Table[Binomial[9 n, 3 n]/(6 n + 1), {n, 0, 20}]

a(n) = A001764(3*n) = A047749(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/9,2/9,4/9,5/9,7/9,8/9; 1/3,1/2,2/3,5/6,7/6; 19683*x/64).
a(n) ~ 3^(9*n-1)/(sqrt(Pi)*4^(3*n+1)*n^(3/2)). (End)
D-finite with recurrence 8*(6*n + 5)*(2*n + 1)*(n + 1)*(3*n + 2)*(3*n + 1)*(6*n + 7)*a(n + 1) = 3*(9*n + 8)*(9*n + 7)*(9*n + 5)*(9*n + 4)*(9*n + 2)*(9*n + 1)*a(n). - Robert Israel, Feb 15 2021

A337292 a(n) = 4binomial(5n,n)/(5*n-1).

Original entry on oeis.org

5, 20, 130, 1020, 8855, 81900, 791120, 7887660, 80560285, 838553320, 8863227100, 94871786100, 1026317094705, 11203116342560, 123243929011680, 1364973221797900, 15207477517956825, 170321264840835900, 1916512328325665070, 21655893753689280120

Offset: 1

Lucas A. Brown, Aug 21 2020

a(n) is the number of lattice paths from (0,0) to (4n,n) using only the steps (1,0) and (0,1) and whose only lattice points on the line y = x/4 are the path's endpoints.

Cf. A002294, A118971, A337291.

Array[4 Binomial[5 #, #]/(5 # - 1) &, 20] (* Michael De Vlieger, Aug 21 2020 *)

a(n) = {4*binomial(5*n,n)/(5*n-1)} \\ Andrew Howroyd, Aug 21 2020

a(n) = 5*A118971(n-1).
G.f.: 5*x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.
D-finite with recurrence 8*n*(4*n-3)*(2*n-1)*(4*n-1)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-6)*a(n-1)=0., a(0)=1. - R. J. Mathar, Jan 26 2025
G.f.: -4*4F3(-1/5,1/5,2/5,3/5; 1/4,1/2,3/4; 3125*x/256) . - R. J. Mathar, Aug 10 2025

A349582 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^3 A(x)^5.

Original entry on oeis.org

1, 3, 13, 85, 733, 7292, 78267, 880250, 10226237, 121713373, 1476272394, 18180126906, 226704989103, 2856790765238, 36321840773980, 465362291912648, 6002272018481901, 77873186277771107, 1015583616140910999, 13306207249869273003, 175064043975233981626

Offset: 0

Ilya Gutkovskiy, Nov 22 2021

nonn

Second binomial transform of A002294.

Cf. A002294, A064613, A346647, A346762, A349581, A349584, A349590, A349591.

nmax = 20; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^3 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[5 k, k] 2^(n - k)/(4 k + 1), {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(5*k,k)*2^(n-k)/(4*k+1)); \\ Michel Marcus, Nov 23 2021

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*k,k) * 2^(n-k) / (4*k+1).
a(n) = 2^n*F([1/5, 2/5, 3/5, 4/5, -n], [1/2, 3/4, 1, 5/4], -5^5/2^9), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 3637^(n + 3/2) / (78125 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Nov 26 2021

A366037 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 5 * A(x)).

Original entry on oeis.org

0, 1, 10, 160, 3110, 67155, 1548526, 37346040, 930513870, 23765376580, 618871054120, 16370119905880, 438628647940730, 11880264846822610, 324739360804852980, 8946782070689651280, 248184394985913218910, 6926162613387923126700, 194320992885495965332600, 5477763483026946993808960, 155070883903415687652796120

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for 4-dimensional figurate numbers A002419 (with signs).

Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002294, A002419, A064089, A365668, A365755, A365818, A366017, A366035, A366036.

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

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

a(n) ~ sqrt((5168 - 869*sqrt(34)) / (17*Pi)) * (22 - sqrt(34))^(5*n) / (2 * n^(3/2) * 3^(3*n + 3/2) * 5^(4*n + 1) * (11*sqrt(34) - 62)^n). - Vaclav Kotesovec, Sep 27 2023

A377526 E.g.f. satisfies A(x) = 1 + xexp(x)A(x)^5.

Original entry on oeis.org

1, 1, 12, 273, 9604, 460105, 27966126, 2062219117, 178897527768, 17853102321489, 2014988044093210, 253792946798597701, 35290880970687039732, 5370055269772474994713, 887591963820839894529654, 158357028389450319651183165, 30332317748593431632078480176, 6208425034878692992471996557217

Offset: 0

Seiichi Manyama, Oct 30 2024

nonn

In general, for k > 1, if e.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^k, then a(n) ~ sqrt(k*(1 + LambertW((k-1)^(k-1)/k^k))) * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW((k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Nov 11 2024

Cf. A006153, A295238, A364983, A364987.

Cf. A002294.

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k, k)/((4*k+1)*(n-k)!));

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002294(k)/(n-k)!.

a(n) ~ sqrt(5*(1 + LambertW(256/3125))) * n^(n-1) / (8 * exp(n) * LambertW(256/3125)^n). - Vaclav Kotesovec, Nov 11 2024

A378920 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 + xA(x)^2).

Original entry on oeis.org

1, 1, 5, 38, 339, 3308, 34191, 367844, 4076112, 46204209, 533239820, 6244542391, 74016115926, 886276231388, 10704869669941, 130271156244371, 1595708949486866, 19658780721376791, 243429900033986385, 3028086095940468087, 37821457123957529163, 474145963420441744445

Offset: 0

Seiichi Manyama, Dec 11 2024

nonn

Cf. A002294, A349362, A364765, A365218, A378892, A378919.

Cf. A000108, A219537, A291534, A365225.

a(n, r=1, s=-1, t=6, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^5/(1 + x*A(x)^2)).

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r).

A235536 a(n) = binomial(8n, 2n) / (6*n + 1).

Original entry on oeis.org

1, 4, 140, 7084, 420732, 27343888, 1882933364, 134993766600, 9969937491420, 753310723010608, 57956002331347120, 4524678117939182220, 357557785658996609700, 28545588568201512137904, 2298872717007844035521848, 186533392975795702301759056

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=6, k=2 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.
First bisection of A002293.
Also, the sequence is between A002296 and A235535.

Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), this sequence (l=6, k=2), A187357 (l=3, k=3), A235535 (l=6, k=3).

l:=6; k:=2; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* where l is divisible by all the prime factors of k */

Table[Binomial[8 n, 2 n]/(6 n + 1), {n, 0, 20}]

a(n) = A124753(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/8,1/4,3/8,5/8,3/4,7/8; 1/3,1/2,2/3,5/6,7/6; 65536*x/729).
a(n) ~ 2^(16*n-1)/(sqrt(Pi)*3^(6*n+3/2)*n^(3/2)). (End)

cp's OEIS Frontend

A386699 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n,k).

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A206290 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/(1 + x^k) ).

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A235534 a(n) = binomial(6*n, 2*n) / (4*n + 1).

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A235535 a(n) = binomial(9*n, 3*n) / (6*n + 1).

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A337292 a(n) = 4*binomial(5*n,n)/(5*n-1).

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A349582 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^3 * A(x)^5.

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A366037 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 5 * A(x)).

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A377526 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^5.

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A235534 a(n) = binomial(6n, 2n) / (4*n + 1).

A235535 a(n) = binomial(9n, 3n) / (6*n + 1).

A337292 a(n) = 4binomial(5n,n)/(5*n-1).

A349582 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^3 A(x)^5.

A377526 E.g.f. satisfies A(x) = 1 + xexp(x)A(x)^5.

A378920 G.f. A(x) satisfies A(x) = 1 + xA(x)^6/(1 + xA(x)^2).

A235536 a(n) = binomial(8n, 2n) / (6*n + 1).