A004355 -id:A004355 - cp's OEIS frontend

A226705 G.f.: 1 / (1 + 12xG(x)^4 - 16xG^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

1, 4, 48, 600, 7856, 105684, 1447392, 20075416, 281086416, 3964453368, 56240518128, 801624722232, 11470976280960, 164691196943212, 2371222443727584, 34224696393237360, 495036708728067088, 7173892793100898728, 104135761805147016096, 1513892435551302963792

Offset: 0

Paul D. Hanna, Jun 15 2013

nonn

			G.f.: A(x) = 1 + 4*x + 48*x^2 + 600*x^3 + 7856*x^4 + 105684*x^5 +...
A related series is G(x) = 1 + x*G(x)^6, where
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
G(x)^4 = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +...
G(x)^5 = 1 + 5*x + 40*x^2 + 385*x^3 + 4095*x^4 + 46376*x^5 +...
such that A(x) = 1/(1 + 12*x*G(x)^4 - 16*x*G^5).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A226706, A183160, A147855, A002295.
Cf. A226733, A226751.
Cf. A004355, A079679.

Table[Sum[Binomial[3*n+2*k,n-k]*Binomial[3*n-2*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 16 2013 *)

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^6+x*O(x^n)); polcoeff(1/(1+12*x*G^4-16*x*G^5), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(G=1+x); for(i=0, n,G=1+x*G^6+x*O(x^n)); polcoeff(1/(1-4*x*G^4-16*x^2*G^10), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(6*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(3*n +2*k, n-k)*binomial(3*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(6*n +2*k, n-k)*binomial(-2*k, k))}
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=0..n} C(2*k, n-k) * C(6*n-2*k, k).
a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(5*n-2*k, k).
a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(4*n-2*k, k).
a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(3*n-2*k, k).
a(n) = Sum_{k=0..n} C(4*n+2*k, n-k) * C(2*n-2*k, k).
a(n) = Sum_{k=0..n} C(5*n+2*k, n-k) * C(n-2*k, k).
a(n) = Sum_{k=0..n} C(6*n+2*k, n-k) * C(-2*k, k).
Self-convolution of A226706.
G.f.: 1 / (1 - 4*x*G(x)^4 - 16*x^2*G(x)^10) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
a(n) ~ 2^(6*n-2)*3^(6*n+3/2)/(5^(5*n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 16 2013
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = [x^n] 1/((1+2*x) * (1-x)^(5*n+1)).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(6*n+1,k).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).
G.f.: G(x)^2/((-2+3*G(x)) * (6-5*G(x))) where G(x) = 1+x*G(x)^6 is the g.f. of A002295. (End)
G.f.: B(x)^2/(1 + 4*(B(x)-1)/3), where B(x) is the g.f. of A004355. - Seiichi Manyama, Aug 15 2025

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

A090969 a(n) = 1/Integral_{x=0..1} (x^5 - x^6)^n.

Original entry on oeis.org

1, 42, 858, 15504, 265650, 4417686, 72068304, 1160068104, 18490100706, 292486494300, 4599035681526, 71963547329856, 1121519754006288, 17419158268943970, 269767427275060200, 4167406330765934256

Offset: 0

Al Hakanson (hawkuu(AT)excite.com), Feb 29 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 0..845

Cf. A002457, A004355, A016921, A090816, A090957.

[Factorial(6*n+1)/(Factorial(n)*Factorial(5*n)): n in [0..20]]; // G. C. Greubel, Feb 03 2019

seq(factorial(6*n+1)/(factorial(n)*factorial(5*n)), n = 0 .. 16); # Emeric Deutsch, Jun 29 2009

Table[1/Beta[5*n+1, n+1], {n,0,20}] (* G. C. Greubel, Feb 03 2019 *)

vector(20, n, n--; (6*n+1)!/(n!*(5*n)!)) \\ G. C. Greubel, Feb 03 2019

[1/beta(5*n+1,n+1) for n in range(20)] # G. C. Greubel, Feb 03 2019

a(n) = A016921(n)*A004355(n). - R. J. Mathar, Jun 21 2009
a(n) = 1/B(5*n+1,n+1) = (6*n+1)!/(n! * (5*n)!), where B(p,q) is Euler's beta function (basically identical with R. J. Mathar's comment). - Emeric Deutsch, Jun 29 2009
a(n) ~ 2^(6*n+1) * 3^(6*n+3/2) * sqrt(n) / (sqrt(Pi) * 5^(5*n+1/2)). - Vaclav Kotesovec, Aug 15 2017

Extended by Emeric Deutsch, Jun 29 2009

A023852 Sum of exponents in prime-power factorization of binomial(6n, n).

Original entry on oeis.org

0, 2, 3, 6, 5, 7, 9, 9, 8, 10, 10, 14, 13, 12, 15, 16, 15, 16, 16, 19, 16, 18, 22, 24, 22, 23, 21, 23, 22, 22, 24, 25, 25, 24, 25, 28, 25, 27, 28, 28, 26, 31, 30, 38, 36, 33, 38, 38, 37, 35, 36, 38, 33, 33, 37, 38, 36, 39, 37, 43, 39, 41, 43, 46, 42, 40, 41, 45, 46, 43, 47, 46, 45, 46, 47, 49, 47, 45

Offset: 0

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A001222, A004355, A022559.

Cf. A023853, A023854.

with(numtheory):a:=proc(n) if n=0 then 0 else bigomega(binomial(6*n,n)) fi end: seq(a(n), n=0..77); # Zerinvary Lajos, Apr 11 2008

Table[PrimeOmega[Binomial[6 n, n]], {n, 0, 77}] (* Ivan Neretin, Nov 09 2017 *)

a(n) = bigomega(binomial(6*n, n)); \\ Michel Marcus, Nov 10 2017

a(n) = A001222(A004355(n)). - Michel Marcus, Nov 10 2017

a(n) = A022559(6*n) - A022559(5*n) - A022559(n). - Amiram Eldar, Jun 11 2025

a(0)=0 inserted by Amiram Eldar, Jun 11 2025

A169959 a(n) = binomial(10*n, n).

Original entry on oeis.org

1, 10, 190, 4060, 91390, 2118760, 50063860, 1198774720, 28987537150, 706252528630, 17310309456440, 426342151127100, 10542859559688820, 261594860525768000, 6509613950241656640, 162392216278033616560, 4059949873964357469950, 101696990867999141755140

Offset: 0

N. J. A. Sloane, Aug 07 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..115

binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).

[Binomial(10*n, n): n in [0..50]]; // Vincenzo Librandi, Apr 21 2011

A169959[n_] := Binomial[10*n, n]; Array[A169959, 20, 0] (* Paolo Xausa, Aug 20 2025 *)

a(n) = C(10*n-1, n-1)*C(100*n^2, 2)/(3*n*C(10*n+1, 3)), n > 0. - Gary Detlefs, Jan 02 2014
From Peter Bala, Feb 21 2022: (Start)
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 9*A(x))^9 +  (10^10)*x*A(x)^10 = 0.
Sum_{n >= 1} a(n)*( x*(9*x + 10)^9/(10^10*(1 + x)^10) )^n = x. (End)

A169960 a(n) = binomial(11*n,n).

Original entry on oeis.org

1, 11, 231, 5456, 135751, 3478761, 90858768, 2404808340, 64276915527, 1731030945644, 46897636623981, 1276749965026536, 34898565177533200, 957150015393611193, 26327386978706181060, 725971390105457325456, 20062118235172477959495, 555476984964439251664995

Offset: 0

N. J. A. Sloane, Aug 07 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).

[Binomial(11*n, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

Table[Binomial[11 n, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)

a(n) = C(11*n-1,n-1)*C(121*n^2,2)/(3*n*C(11*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014
From Peter Bala, Feb 21 2022: (Start)
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 10*A(x))^10 + (11^11)*x*A(x)^11 = 0.
Sum_{n >= 1} a(n)*( x*(10*x + 11)^10/(11^11*(1 + x)^11) )^n = x. (End)

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

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

A380607 a(0) = 1, a(n) = 5binomial(6(n-1),n-1), for n > 0.

Original entry on oeis.org

1, 5, 30, 330, 4080, 53130, 712530, 9738960, 134891640, 1886744970, 26589681300, 376970137830, 5370413979840, 76816421507280, 1102478371452150, 15868672192650600, 228978369822304080, 3311260421942706570

Offset: 0

Karol A. Penson, Jan 28 2025

nonn

Cf. A004355.

CoefficientList[Series[5*z*HypergeometricPFQ[{1/6, 1/3, 1/2, 2/3, 5/6}, {1/5, 2/5, 3/5, 4/5}, (6^6*z)/5^5] + 1,{z,0,17}],z] (* Stefano Spezia, Jan 28 2025 *)

a(n) = 0^n+binomial(6*(n-1),n-1)*5 \\ Thomas Scheuerle, Jan 29 2025

a(n) = 5*A004355(n-1), for n>=1.
G.f.: h(z) = 5*z*hypergeom([1/6, 1/3, 1/2, 2/3, 5/6], [1/5, 2/5, 3/5, 4/5], (6^6*z)/5^5) + 1
satisfies: 15625*z^6 - 75000*z^5 + 140625*z^4 - 125000*z^3 + 46875*z^2 + 46656*z - 3125 + (75000*z^5 - 281250*z^4 + 375000*z^3 - 187500*z^2 - 279936*z + 18750)*h(z) + (140625*z^4 - 375000*z^3 + 281250*z^2 + 699840*z - 46875)*h(z)^2 + (125000*z^3 - 187500*z^2 - 933120*z + 62500)*h(z)^3 + (46875*z^2 + 699840*z - 46875)*h(z)^4 + (-279936*z + 18750)*h(z)^5 + (46656*z - 3125)*h(z)^6 = 0.
a(n) = Integral_{x=0..sup} x^n*W(x), where sup = 6^6/5^5, with W(x) = (5^10)*sqrt(15)/((6^12)*sqrt(Pi) )*MeijerG([[],[-1,-9/5,-8/5,-7/5,-6/5]],[[-11/6,-5/3,-3/2,-4/3,-7/6],[]],x/(6^6/5^5)), n>0. In W(x) MeijerG is the Meijer G-function in Maple notation, which can be represented as the sum of five  generalized hypergeometric functions of type 5F4. This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem on x = (0, sup). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is U-shaped, is singular at x = 0, with singularity x^(-1/6), and is singular at x = sup. W(x) has a minimum at x around x=11.

cp's OEIS Frontend

A226705 G.f.: 1 / (1 + 12*x*G(x)^4 - 16*x*G^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

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

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A090969 a(n) = 1/Integral_{x=0..1} (x^5 - x^6)^n.

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A023852 Sum of exponents in prime-power factorization of binomial(6n, n).

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A169959 a(n) = binomial(10*n, n).

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A169960 a(n) = binomial(11*n,n).

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

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A226705 G.f.: 1 / (1 + 12xG(x)^4 - 16xG^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

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

A380607 a(0) = 1, a(n) = 5binomial(6(n-1),n-1), for n > 0.