A007556 -id:A007556 - cp's OEIS frontend

A346582 a(n) = (1/(8n)) Sum_{d|n} mu(n/d) * binomial(8*d,d).

1, 7, 84, 1120, 16450, 255612, 4141382, 69158272, 1182125043, 20581143150, 363704640475, 6506965023168, 117626432708863, 2145180354493274, 39421026305266125, 729242353100281344, 13568988503585900647, 253785064585174078869, 4768543107831461199896, 89970814565326816488000

Offset: 1

Ilya Gutkovskiy, Jul 24 2021

nonn

Inverse Euler transform of A007556.

Moebius transform of A261501.

Cf. A004381, A007556, A008683, A022553, A261501, A346577, A346578, A346579, A346580, A346581.

Table[(1/(8 n)) Sum[MoebiusMu[n/d] Binomial[8 d, d], {d, Divisors[n]}], {n, 20}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(8*d,d))/(8*n); \\ Michel Marcus, Jul 24 2021

A346684 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8k,k) / (7k + 1).

Original entry on oeis.org

1, 0, 8, 84, 1156, 17122, 268262, 4370086, 73281938, 1256608767, 21933420953, 388400019583, 6960642974905, 126008367913375, 2300862338502425, 42326714610861679, 783717720798538121, 14594469249932149279, 273161824453612674593, 5135931850101477641707

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

nonn

In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m + (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Seiichi Manyama, Table of n, a(n) for n = 0..768

Cf. A007556, A032357, A188678, A346668, A346672.

Cf. A346680, A346681, A346682, A346683.

Table[Sum[(-1)^(n - k) Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^7 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(8*k, k)/(7*k + 1)); \\ Michel Marcus, Jul 29 2021

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^7 * A(x)^8.

a(n) ~ 2^(24*n + 25) / (17600759 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(nk,k) / ((n-1)k + 1).

Original entry on oeis.org

1, 2, 5, 25, 257, 4361, 104425, 3241316, 123865313, 5628753361, 296671566941, 17798975341467, 1197924420178381, 89394126594968755, 7326377073291002147, 654215578855903951141, 63225054646397348577601, 6575059243843086616460321, 732138834180570978286488133

Offset: 0

Vaclav Kotesovec, Nov 23 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..338

Cf. A007317, A007556, A188687, A346646, A346647, A346648, A346649, A346650.

Cf. A378326.

Table[Sum[Binomial[n, k] Binomial[n*k, k]/((n-1)*k + 1), {k, 0, n}], {n, 0, 20}]

a(n) ~ exp(n + exp(-1) - 1/2) * n^(n - 5/2) / sqrt(2*Pi).

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

A346939 a(n) = Sum_{d|n} mu(n/d) * binomial(8d,d) / (7d+1).

Original entry on oeis.org

1, 7, 91, 1232, 18277, 285285, 4638347, 77650784, 1329890613, 23190011435, 410333440535, 7349042707872, 132969010888279, 2426870701777445, 44627576949345735, 826044435331747776, 15378186970730687399, 287756293702214647875, 5409093674555090316299, 102094541350713952736608

Offset: 1

Ilya Gutkovskiy, Aug 08 2021

nonn

Moebius transform of A007556.

Cf. A002996, A007556, A008683, A346925, A346935, A346936, A346937, A346938.

Table[Sum[MoebiusMu[n/d] Binomial[8 d, d]/(7 d + 1), {d, Divisors[n]}], {n, 20}]

A349591 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^6 A(x)^8.

Original entry on oeis.org

1, 3, 16, 160, 2216, 35110, 596016, 10573748, 193586424, 3629709697, 69342483276, 1344897261828, 26411276859800, 524117511080056, 10493756451964088, 211719733855698808, 4300202981875132408, 87854045612854431128, 1804215079309443709632

Offset: 0

Ilya Gutkovskiy, Nov 22 2021

nonn

Second binomial transform of A007556.

Cf. A007556, A064613, A346650, A346762, A349581, A349582, A349584, A349590.

nmax = 18; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^6 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[8 k, k] 2^(n - k)/(7 k + 1), {k, 0, n}], {n, 0, 18}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(8*k,k)*2^(n-k)/(7*k+1)); \\ Michel Marcus, Nov 23 2021

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(8*k,k) * 2^(n-k) / (7*k+1).
a(n) = 2^n*F([1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, -n], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7], -2^23/7^7), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 2^(n - 67/2) * 9212151^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 26 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)

A383965 Self-convolution square-root of A004381, where A004381(n) = binomial(8*n,n).

Original entry on oeis.org

1, 4, 52, 804, 13412, 233548, 4180932, 76307228, 1412731844, 26443784224, 499310856828, 9494966722696, 181620437132820, 3491268491768400, 67396227598309788, 1305787014634864584, 25380012805871145604, 494684878753394992992, 9665968233663380580256, 189289570996914582016788

Offset: 0

Vaclav Kotesovec, Jun 06 2025

nonn

In general, for m > 1, if Sum_{k=0..n} a(k)*a(n-k) = binomial(m*n,n), then a(n) ~ m^(m*n + 1/4) / (2^(1/4) * Gamma(1/4) * (m-1)^((m-1)*n + 1/4) * n^(3/4)).

Cf. A004381, A007556, A208977, A384695.

a:= proc(n) option remember; `if`(n=0, 1,
      (binomial(8*n, n)-add(a(j)*a(n-j), j=1..n-1))/2)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 06 2025

nmax = 20; self = ConstantArray[0, nmax + 1]; self[[1]] = 1; self[[2]] = 4; Do[self[[k+1]] = (Binomial[8*k, k] - Sum[self[[j+1]]*self[[k - j + 1]], {j, 1, k-1}]) / (2*self[[1]]);, {k, 2, nmax}]; self

a(n) ~ 2^(24*n + 1/2) / (Gamma(1/4) * 7^(7*n + 1/4) * n^(3/4)).
From Seiichi Manyama, Aug 16 2025: (Start)
Sum_{k=0..n} a(k) * a(n-k) = A004381(n).
G.f.: 1/sqrt(1 - 8*x*g^7) where g = 1+x*g^8 is the g.f. of A007556.
G.f.: sqrt( g/(8-7*g) ) where g = 1+x*g^8 is the g.f. of A007556. (End)

A059967 Number of 9-ary trees.

Original entry on oeis.org

1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, 1416298046436, 28748759731965, 589546754316126, 12195537924351375, 254184908607118800, 5332692942907262361, 112524941404978156215

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Mar 05 2001

nonn

S. Heubach, N. Y. Li and T. Mansour, Staircase tilings and k-Catalan structures, Discrete Math., 308 (2008), 5954-5964.
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Related algebraic sequences concerning trees: strictly k-ary trees (A000108: s=x+s^2, A001263: s=(x, y)+(x, s)+(s, y)+(s, s))), (A001764: s=x+s^3), (A002293: s=x+s^4), (A002294: s=x+s^5), (A002295: s=x+s^6), (A002296: s=x+s^7), (A007556: s=x+s^8), at most k-ary trees (A001006: s=x+xs+xs^2), (A036765-A036769, s=x+xs^2....+xs^k, k=3, 4, 5, 6, 7).

with(combinat): for n from 1 to 40 do printf(`%d,`,binomial(9*n,n)/((9-1)*n+1)) od:

G.f. A(x) satisfies: A = x + A^9.

a(n) = C(k*n, n)/((k-1)*n+1), k=9.

More terms from James Sellers, Mar 15 2001

cp's OEIS Frontend

A346582 a(n) = (1/(8*n)) * Sum_{d|n} mu(n/d) * binomial(8*d,d).

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A346684 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8*k,k) / (7*k + 1).

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A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k) / ((n-1)*k + 1).

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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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A346939 a(n) = Sum_{d|n} mu(n/d) * binomial(8*d,d) / (7*d+1).

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A349591 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^6 * A(x)^8.

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

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A346582 a(n) = (1/(8n)) Sum_{d|n} mu(n/d) * binomial(8*d,d).

A346684 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8k,k) / (7k + 1).

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(nk,k) / ((n-1)k + 1).

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

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

A346939 a(n) = Sum_{d|n} mu(n/d) * binomial(8d,d) / (7d+1).

A349591 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^6 A(x)^8.

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