A002295 -id:A002295 - cp's OEIS frontend

A346937 a(n) = Sum_{d|n} mu(n/d) * binomial(6d,d) / (5d+1).

1, 5, 50, 500, 5480, 62776, 749397, 9203128, 115607259, 1478308780, 19180049927, 251857056364, 3340843549854, 44700484300317, 602574657421585, 8175951649914160, 111572030260242089, 1530312970224714489, 21085148778264281864, 291705220703240850760, 4050527291832419432577

Offset: 1

Ilya Gutkovskiy, Aug 08 2021

nonn

Moebius transform of A002295.

Cf. A002295, A002996, A008683, A346925, A346935, A346936, A346938, A346939.

Table[Sum[MoebiusMu[n/d] Binomial[6 d, d]/(5 d + 1), {d, Divisors[n]}], {n, 21}]

A349584 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^4 A(x)^6.

Original entry on oeis.org

1, 3, 14, 107, 1106, 13173, 168820, 2264298, 31356818, 444803666, 6429510234, 94356870748, 1402149248128, 21055387206719, 319007902203196, 4870481885025752, 74858763620576738, 1157339247553310574, 17985974981514604660, 280813589679135551721

Offset: 0

Ilya Gutkovskiy, Nov 22 2021

nonn

Second binomial transform of A002295.

Cf. A002295, A064613, A346648, A346762, A349581, A349582, A349590, A349591.

nmax = 19; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^4 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[6 k, k] 2^(n - k)/(5 k + 1), {k, 0, n}], {n, 0, 19}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(6*k,k)*2^(n-k)/(5*k+1)); \\ Michel Marcus, Nov 23 2021

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*k,k) * 2^(n-k) / (5*k+1).
a(n) = 2^n*F([1/6, 1/3, 1/2, 2/3, 5/6, -n], [2/5, 3/5, 4/5, 1, 6/5], -3^6*(2/5)^5), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 2^(n - 15/2) * 26453^(n + 3/2) / (6561 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 26 2021

A365194 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^6).

Original entry on oeis.org

1, 1, 6, 52, 529, 5889, 69462, 853013, 10791018, 139659604, 1840435530, 24611295075, 333132371248, 4555465710569, 62839303262352, 873363902976309, 12218178082489873, 171918448407833112, 2431415226089290680, 34544425914499450493, 492807213597429920649

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A243667, A349332, A364748, A365192, A365193.

Cf. A219537, A271469, A364765.

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

a(n) = Sum_{k=0..n} binomial(6*n-k+1,k) * binomial(n-1,n-k)/(6*n-k+1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(5*n+2*k+1,k) * binomial(n-1,n-k)/(5*n+2*k+1).
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(6*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Dec 26 2024

A052795 a(n) = (6n)!/(5n+1)!.

Original entry on oeis.org

1, 1, 12, 306, 12144, 657720, 45239040, 3776965920, 371090522880, 41951580652800, 5364506808460800, 765606216965990400, 120639963305775513600, 20803502274492921984000, 3896911902445736638464000, 787971434323820421362688000, 171063718698166603304067072000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Old name was: A simple grammar.

Seiichi Manyama, Table of n, a(n) for n = 0..310
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 752

Cf. A001761, A001763, A365340, A365341.

Cf. A000142, A002295.

spec := [S,{B=Prod(Z,S,S,S,S,S),S=Sequence(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20); # end of program
seq((6*n)!/(5*n+1)!, n=0..20);  # Mark van Hoeij, May 29 2013

a(n) = (6*n)!/(5*n+1)!; \\ Joerg Arndt, May 29 2013

from sympy import ff
def A052795(n): return ff(6*n,n-1) # Chai Wah Wu, Sep 01 2023

E.g.f.: RootOf(-_Z+_Z^6*x+1).
D-finite Recurrence: {a(1)=1, a(2)=12, (-720-9864*n-48600*n^2-110160*n^3-116640*n^4-46656*n^5)*a(n)+(3125*n^4+9375*n^3+10000*n^2+4500*n+720)*a(n+1), a(6)=45239040, a(3)=306, a(4)=12144, a(5)=657720}.
1/25*3^(1/2)*(5+5^(1/2))^(1/2)*(5-5^(1/2))^(1/2)*Pi^(1/2) *GAMMA(2*n+37/3) *GAMMA(2*n+38/3)/GAMMA(n+34/5)/GAMMA(n+33/5)/GAMMA(n+32/5) /GAMMA(n+36/5) *GAMMA(n+13/2)*3125^(-6-n)*2916^(n+6).
a(n) = (6*n)!/(5*n+1)!. - Mark van Hoeij, May 29 2013
E.g.f.: exp( 1/6 * Sum_{k>=1} binomial(6*k,k) * x^k/k ). - Seiichi Manyama, Feb 08 2024
a(n) = A000142(n)*A002295(n). - Alois P. Heinz, Feb 08 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^5).
a(n) = Sum_{k=0..n} (5*n+1)^(k-1) * |Stirling1(n,k)|. (End)

New name using Mark van Hoeij's formula from Joerg Arndt, Feb 18 2019

Accidentally removed a(0) reinserted by Georg Fischer, May 09 2021

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)

A381989 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 19, 514, 22621, 1369546, 105616639, 9901346554, 1093292035609, 138977379784882, 19990424969236171, 3209995501651871890, 569216406245186726965, 110476637766622355475898, 23294266811686640511534199, 5302371488162151660366545866, 1295920217231693678343467474353

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A381987, A381988.

Cf. A002293, A002295, A382001.

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

Let F(x) be the e.g.f. of A382001. F(x) = B(x*A(x)^2) = exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * (x*A(x)^2)^k/k ).

a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k) * A002295(k)/(n-k)!.

A384943 G.f. A(x) satisfies A(x) = 1 + x/A(-x*A(x))^6.

Original entry on oeis.org

1, 1, 6, -9, -244, -39, 11262, 36971, -268890, -3724293, -24899558, 159971919, 3851093928, 9663394063, -197371002600, -2108992348026, -9447769941412, 111942512192787, 2253965670439788, 7917705821761592, -100488750700889250, -1520857626228210483

Offset: 0

Seiichi Manyama, Jun 13 2025

sign

Column k=1 of A384946.
Cf. A002295, A213105.
Cf. A000012, A384894, A384896, A384941, A384942.

a(n, k=-1) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+2*j+k-1, j-1)*a(n-j, 6*j)/j));

See A384946.

A206289 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x*(1 - x^k) ).

Original entry on oeis.org

1, 1, 2, 4, 10, 25, 73, 214, 679, 2189, 7331, 24867, 86269, 302144, 1072621, 3837768, 13853674, 50319789, 183941789, 675731105, 2494370326, 9244865453, 34394851701, 128390336942, 480749791772, 1805161153783, 6795744287172, 25643914891284, 96980809856731

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Compare to the g.f. of partitions of n into distinct parts (A000009): Sum_{n>=0} Product_{k=1..n} x*(1 + x^k).

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 73*x^6 + 214*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 + 2*x^3 + 5*x^4 + 14*x^5 +...+ A000108(n)*x^(n+1) +...
G_2(x) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 +...+ A001764(n)*x^(2*n+1) +...
G_3(x) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 +...+ A002293(n)*x^(3*n+1) +...
G_4(x) = x + x^5 + 5*x^9 + 35*x^13 + 285*x^17 +...+ A002294(n)*x^(4*n+1) +...
G_5(x) = x + x^6 + 6*x^11 + 51*x^16 + 506*x^21 +...+ A002295(n)*x^(5*n+1) +...
G_6(x) = x + x^7 + 7*x^13 + 70*x^19 + 819*x^25 +...+ A002296(n)*x^(6*n+1) +...
Note that G_n(x) = x + x*G_n(x)^(n+1).

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Cf. A206290, A194560.

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x*(1-x^k+x*O(x^n))))),n)}
for(n=0,35,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+1),
(3) G_n(x) = Sum_{k>=0} binomial(n*k+k+1, k) * x^(n*k+1) / (n*k+k+1).
a(n) ~ c * 4^n / n^(3/2), where c = 0.19197348199... . - Vaclav Kotesovec, Nov 06 2014

A385497 a(n) = Sum_{k=0..n} binomial(6*n+1,k).

Original entry on oeis.org

1, 8, 92, 1160, 15276, 206368, 2835200, 39419864, 553000876, 7811733392, 110962066532, 1583318009160, 22677731944032, 325849065291056, 4694837606889424, 67803714186207280, 981265566082447276, 14227018304102548368, 206608052310739404392, 3004777578508008253808

Offset: 0

Seiichi Manyama, Aug 17 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A000302, A160906, A386811, A386812.
Cf. A004355, A079590, A079679, A226705.
Cf. A002295, A385719.

[&+[Binomial(6*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 18 2025

Table[Sum[Binomial[6*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 18 2025 *)

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

a(n) = [x^n] (1+x)^(6*n+1)/(1-x).
a(n) = [x^n] 1/((1-x)^(5*n+1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(6*n-k,n-k).
G.f.: 1/(1 - 4*x*g^4*(3-g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: g^2/((2-g) * (6-5*g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: B(x)^2/(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.
a(n) ~ 2^(6*n-1) * 3^(6*n + 3/2) / (sqrt(Pi*n) * 5^(5*n + 1/2)). - Vaclav Kotesovec, Aug 19 2025
D-finite with recurrence +5*n*(5*n-3) *(25275337086729240289198339046875*n +471647298106881091699147254457046) *(5*n-1)*(5*n-4)*(5*n-2)*a(n) +(78985428396028875903744809521484375*n^6 -559942234844855804767211877804090453801*n^5 +3587636672285250929619857349305543417315*n^4 -10153151347942687598200945831585305558855*n^3 +14794114656715293872778407292185015920550*n^2 -10846691360081598422810600143797325763664*n +3179147242764665659301361496311050364480)*a(n-1) +40*(916451705547792050816664342989042382392*n^6 -15754440652132350078674083937326518806004*n^5 +117614110896134855700514819789186651267682*n^4 -471111363407608954402735569277858473721059*n^3 +1053743992048348087929158710510276422876431*n^2 -1242809524683997363700671579060256757555078*n +603414490131980309336751304501155726403152) *a(n-2) +3072*(-950768355029313182341332806167821761828*n^6 +17097100921628721474237101055297828968024*n^5 -128090998271831890487248970509140383514230*n^4 +509544263618626898681417576914870842148685*n^3 -1132270964907780344616429736070172799129247*n^2 +1330655887974191637410201798934319046990726*n -645481184978535641217111809931780144149880) *a(n-3) +884736*(3*n-11) *(6*n-17) *(61801507754400081418308631750717123*n -123657551673181017806623428016627104) *(6*n-19)*(3*n-10)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Aug 26 2025

A385719 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.

Original entry on oeis.org

1, 4, 38, 428, 5204, 66104, 863840, 11515308, 155779966, 2131436392, 29426804398, 409254436452, 5726378247412, 80535621269208, 1137609359823936, 16130112288879248, 229462608491483364, 3273749607191060480, 46826932120849617128, 671341041479214814160, 9644654058165119642624

Offset: 0

Seiichi Manyama, Aug 17 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..850

Cf. A183161, A208977, A387084, A387086.

Cf. A002295, A004355, A385497.

nmax = 20; CoefficientList[Series[Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] - 1)/3], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)

Sum_{k=0..n} a(k) * a(n-k) = A385497(n).
G.f.: 1/sqrt(1 - 4*x*g^4*(3-g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: g/sqrt((2-g) * (6-5*g)) where g = 1+x*g^6 is the g.f. of A002295.
a(n) ~ 2^(6*n - 1/2) * 3^(6*n + 3/4) / (Gamma(1/4) * n^(3/4) * 5^(5*n + 1/4)) * (1 + 7*Gamma(1/4)^2/(48*Pi*sqrt(30*n))). - Vaclav Kotesovec, Aug 20 2025

cp's OEIS Frontend

A346937 a(n) = Sum_{d|n} mu(n/d) * binomial(6*d,d) / (5*d+1).

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A349584 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^4 * A(x)^6.

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A365194 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^6).

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PARI

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A052795 a(n) = (6*n)!/(5*n+1)!.

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Maple

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

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A381989 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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PARI

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A384943 G.f. A(x) satisfies A(x) = 1 + x/A(-x*A(x))^6.

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A206289 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x*(1 - x^k) ).

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A385497 a(n) = Sum_{k=0..n} binomial(6*n+1,k).

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A346937 a(n) = Sum_{d|n} mu(n/d) * binomial(6d,d) / (5d+1).

A349584 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^4 A(x)^6.

A365194 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^6).

A052795 a(n) = (6n)!/(5n+1)!.

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

A381989 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^4 is the g.f. of A002293.