A034835 -id:A034835 - cp's OEIS frontend

A216703 a(n) = Product_{k=1..n} (49 - 7/k).

1, 42, 1911, 89180, 4213755, 200574738, 9594158301, 460519598448, 22162505675310, 1068725273676060, 51619430718553698, 2496503376570051576, 120872371815599997138, 5857661095679076784380, 284096563140435224042430, 13788153197749122873525936

Offset: 0

Michel Lagneau, Sep 16 2012

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Cf. A004988, A004994, A049382, A216702, A220607.

Cf. A034835, A386271, A386272, A386273, A386274.

seq(product(49-7/k, k=1.. n), n=0..20);
seq((7^n/n!)*product(7*k+6, k=0.. n-1), n=0..20);

Table[49^n * Pochhammer[6/7, n] / n!, {n, 0, 15}] (* Amiram Eldar, Aug 17 2025 *)

From Seiichi Manyama, Jul 17 2025: (Start)
G.f.: 1/(1 - 49*x)^(6/7).
a(n) = (-49)^n * binomial(-6/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+6). (End)
From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 49^n * Gamma(n+6/7) / (Gamma(6/7) * Gamma(n+1)).
a(n) ~ c * 49^n / n^(1/7), where c = 1/Gamma(6/7) = 1/A220607 = 0.904349... . (End)

A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

Original entry on oeis.org

1, 4, 40, 480, 6240, 84864, 1188096, 16972800, 246105600, 3609548800, 53421322240, 796463349760, 11946950246400, 180123249868800, 2727580640870400, 41459225741230080, 632253192553758720

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..500
A. Straub, V. H. Moll, and T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10).

Cf. A007696.

Expansion of (1-b^2*x)^(-1/b): A000984 (b=2), A004987 (b=3), this sequence (b=4), A034688 (b=5), A004993 (b=6), A034835 (b=7), A034977 (b=8), A035024 (b=9), A035308 (b=10).

CoefficientList[Series[1/Surd[1-16x,4],{x,0,20}],x] (* Harvey P. Dale, Aug 06 2018 *)

a(n) = (4^n/n!)*A007696(n), n >= 1, a(0) := 1, A007696(n) = (4*n-3)!^4 := Product_{j = 1..n} 4*j - 3.
G.f.: (1 - 16*x)^(-1/4).
D-finite with recurrence: n*a(n) + 4*(-4*n + 3)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-16)^n*binomial(-1/4, n).
a(n) ~ Gamma(3/4)/(sqrt(2)*Pi) * 16^n/n^(3/4).
E.g.f.: hypergeom([1/4], [1], 16*x).
a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-1/4, k)* binomial(-1/4, 2*n - k).
(16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
Sum_{k = 0..n} a(k)*a(n-k) = (4^n)*binomial(2*n, n) = A098430.
Sum_{k = 0..2*n} a(k)*a(2*n-k) = (16^n)*binomial(4*n, 2*n). (End)

A386274 Expansion of 1/(1 - 49*x)^(5/7).

Original entry on oeis.org

1, 35, 1470, 65170, 2965235, 136993857, 6393046660, 300473193020, 14197358370195, 673585780452585, 32062683149543046, 1530264423046372650, 73197648235718158425, 3507856526988647130675, 168377113295455062272400, 8093326579068206659893360, 389491341617657445507367950

Offset: 0

Seiichi Manyama, Jul 17 2025

Harvey P. Dale, Table of n, a(n) for n = 0..592

Cf. A034835, A216703, A386271, A386272, A386273.

CoefficientList[Series[1/(Surd[1-49x,7])^5,{x,0,20}],x] (* Harvey P. Dale, Aug 01 2025 *)

my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(5/7))

a(n) = (-49)^n * binomial(-5/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+5).
a(n) = 7^n * Product_{k=1..n} (7 - 2/k).
D-finite with recurrence n*a(n) +7*(-7*n+2)*a(n-1)=0. - R. J. Mathar, Jul 30 2025

A224881 Expansion of 1/(1 - 16*x)^(1/8).

Original entry on oeis.org

1, 2, 18, 204, 2550, 33660, 460020, 6440280, 91773990, 1325624300, 19354114780, 285033326760, 4227994346940, 63094684869720, 946420273045800, 14259398780556720, 215673406555920390, 3273161111260438860, 49824785804742235980, 760483572809223601800

Offset: 0

Paul D. Hanna, Jul 23 2013

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 204*x^3 + 2550*x^4 + 33660*x^5 + ...
where
A(x)^8 = 1 + 16*x + 256*x^2 + 4096*x^3 + 65536*x^4 + ... + 16^n*x^n + ...
Also,
A(x)^4 = 1 + 8*x + 96*x^2 + 1280*x^3 + 17920*x^4 + 258048*x^5 + ... + 4^n*A000984(n)*x^n + ...
A(x)^2 = 1 + 4*x + 40*x^2 + 480*x^3 + 6240*x^4 + 84864*x^5 + ... + 2^n*A004981(n)*x^n + ...

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

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), this sequence (b=16), A034688 (b=25), A298799 (b=27), A004993 (b=36), A034835 (b=49).

Cf. A301271.

List([0..20],n->(2^n/Factorial(n))*Product([0..n-1],k->8*k+1)); # Muniru A Asiru, Jun 23 2018

seq(coeff(series(1/(1-16*x)^(1/8), x,50),x,n+1),n=0..20); # Muniru A Asiru, Jun 23 2018

CoefficientList[Series[1/(1-16*x)^(1/8), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 24 2013 *)

{a(n)=polcoeff(1/(1-16*x +x*O(x^n))^(1/8),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=(2^n/n!)*prod(k=0,n-1,8*k + 1)}
for(n=0,30,print1(a(n),", "))

a(n) = (2^n/n!) * Product_{k=0..n-1} (8*k + 1).

a(n) ~ 16^n/(GAMMA(1/8)*n^(7/8)). - Vaclav Kotesovec, Jul 24 2013

A298799 Expansion of (1-27*x)^(-1/9).

Original entry on oeis.org

1, 3, 45, 855, 17955, 398601, 9167823, 216098685, 5186368440, 126201632040, 3104560148184, 77049538223112, 1926238455577800, 48452305767226200, 1225151160114148200, 31118839466899364280, 793530406405933789140, 20305042752151835192700

Offset: 0

Seiichi Manyama, Jun 22 2018

nonn

Conjecture: a(p*n) == a(n) (mod p^2) for prime p == 1 (mod 9) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/9 and 1 <= k <= (p-1)/9. Cf. A034171, A004981 and A004982. - Peter Bala, Dec 23 2019

Seiichi Manyama, Table of n, a(n) for n = 0..700

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), A224881 (b=16), A034688 (b=25), this sequence (b=27), A004993 (b=36), A034835 (b=49).

Cf. A305991, A034171, A004981, A004982.

List([0..20],n->(3^n/Factorial(n))*Product([0..n-1],k->9*k+1)); # Muniru A Asiru, Jun 23 2018

seq(coeff(series((1-27*x)^(-1/9), x, n+1), x, n), n=0..20); # Muniru A Asiru, Jun 23 2018
# Alternative:
A298799 := n -> (-27)^n*binomial(-1/9, n):
seq(A298799(n), n=0..17); # Peter Luschny, Dec 26 2019

N=20; x='x+O('x^N); Vec((1-27*x)^(-1/9))

a(n) = 3^n/n! * Product_{k=0..n-1} (9*k + 1) for n > 0.
a(n) ~ 3^(3*n) / (Gamma(1/9) * n^(8/9)). - Vaclav Kotesovec, Jun 23 2018
From Peter Luschny, Dec 26 2019: (Start)
a(n) = (-27)^n*binomial(-1/9, n).
a(n) = n! * [x^n] hypergeom([1/9], [1], 27*x). (End)
D-finite with recurrence: n*a(n) +3*(-9*n+8)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

A386271 Expansion of 1/(1 - 49*x)^(2/7).

Original entry on oeis.org

1, 14, 441, 16464, 662676, 27832392, 1201431588, 52862989872, 2359010923038, 106417603861492, 4842000975697886, 221851681068339504, 10223664969232645476, 473434331652157890504, 22014696421825341908436, 1027352499685182622393680, 48092938891512611510804145

Offset: 0

Seiichi Manyama, Jul 17 2025

Cf. A020918 (k=2, m=7), A020920 (k=2, m=9), A034835 (k=7, m=1), A034977 (k=8, m=1), A035024 (k=9, m=1), A216702 (k=4, m=3), A216703 (k=7, m=6), A354019 (k=6, m=1), this sequence (k=7, m=2), A386272 (k=7, m=3), A386273 (k=7, m=4), A386274 (k=7, m=5).

my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(2/7))

a(n) = (-49)^n * binomial(-2/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+2).
a(n) = 7^n * Product_{k=1..n} (7 - 5/k).
In general, 1/(1 - k^2*x)^(m/k) leads to the D-finite recurrence k*(k*n-k+m)*a(n-1) - n*a(n) = 0. This sequence is case k=7, m=2: (49*n-35)*a(n-1) - n*a(n) = 0. - Georg Fischer, Jul 19 2025

A386272 Expansion of 1/(1 - 49*x)^(3/7).

Original entry on oeis.org

1, 21, 735, 29155, 1224510, 53143734, 2356038874, 106021749330, 4823989594515, 221367522503855, 10227179539678101, 475098976797773601, 22171285583896101380, 1038639455430209672340, 48816054405219854599980, 2300863364299362480145724, 108715793963144877186885459

Offset: 0

Seiichi Manyama, Jul 17 2025

Cf. A034835, A216703, A386271, A386273, A386274.

Cf. A049381, A216702.

my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(3/7))

a(n) = (-49)^n * binomial(-3/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+3).
a(n) = 7^n * Product_{k=1..n} (7 - 4/k).
D-finite with recurrence n*a(n) +7*(-7*n+4)*a(n-1)=0. - R. J. Mathar, Jul 20 2025

A386273 Expansion of 1/(1 - 49*x)^(4/7).

Original entry on oeis.org

1, 28, 1078, 45276, 1980825, 88740960, 4037713680, 185734829280, 8613452707860, 401961126366800, 18851976826602920, 887756726925482960, 41946505347229069860, 1987619022607162079520, 94411903573840198777200, 4494006610114793461794720, 214307940219849213209335710

Offset: 0

Seiichi Manyama, Jul 17 2025

Cf. A034835, A216703, A386271, A386272, A386274.

my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(4/7))

a(n) = (-49)^n * binomial(-4/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+4).
a(n) = 7^n * Product_{k=1..n} (7 - 3/k).
D-finite with recurrence n*a(n) +7*(-7*n+3)*a(n-1)=0. - R. J. Mathar, Jul 20 2025

A034904 Related to sept-factorial numbers A045754.

Original entry on oeis.org

1, 28, 980, 37730, 1531838, 64337196, 2766499428, 121034349975, 5365856182225, 240390356963680, 10861273400995360, 494187939745288880, 22618601857572837200, 1040455685448350511200, 48069052667713793617440, 2229202317465227179008780, 103723472536176158740937940

Offset: 1

Wolfdieter Lang

Convolution of A034835(n-1) with A025752(n), n >= 1.

Michael De Vlieger, Table of n, a(n) for n = 1..594
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Index entries for sequences related to factorial numbers.

Cf. A025752, A034835, A045754, A220086.

CoefficientList[Series[(Power[1-49x, (-7)^-1]-1)/7,{x,0,30}],x] (* Harvey P. Dale, Aug 23 2011 *)

a(n) = 7^(n-1)*A045754(n)/n!, where A045754(n) = (7*n-6)(!^7) = Product_{j=1..n} (7*j-6).
G.f.: (-1+(1-49*x)^(-1/7))/7.
D-finite with recurrence: n*a(n) + 7*(-7*n+6)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 7^(2*n-1) * n^(-6/7) / Gamma(1/7). - Amiram Eldar, Aug 18 2025

A248329 Square array read by antidiagonals downwards: super Patalan numbers of order 7.

Original entry on oeis.org

1, 7, 42, 196, 147, 1911, 6860, 2744, 4459, 89180, 264110, 72030, 62426, 156065, 4213755, 10722866, 2218524, 1310946, 1747928, 5899257, 200574738, 450360372, 75060062, 33647614, 30588740, 55059732, 234003861, 9594158301, 19365495996, 2702162232, 975780806, 672952280, 825895980, 1872030888

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers, A068555, based on Patalan numbers of order 7, A025752.

			T(0..4,0..4) is
  1           7          196        6860       264110
  42          147        2744       72030      2218524
  1911        4459       62426      1310946    33647614
  89180       156065     1747928    30588740   672952280
  4213755     5899257    55059732   825895980  15898497615

Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.

Cf. A068555, A025752, A034835 (first row), A216703 (first column), A248324, A248325, A248326, A248328, A248332.

matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*49^(n+k)*binomial(n-1/7,n+k)) \\ Michel Marcus, Oct 09 2014

T(0,0)=1, T(n,k) = T(n-1,k)*(49*n-7)/(n+k), T(n,k) = T(n,k-1)*(49*k-42)/(n+k).
G.f.: (x/(1-49*x)^(6/7)+y/(1-49*y)^(1/7))/(x+y-49*x*y).
T(n,k) = (-1)^k*49^(n+k)*binomial(n-1/7,n+k).

cp's OEIS Frontend

A216703 a(n) = Product_{k=1..n} (49 - 7/k).

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A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

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A386274 Expansion of 1/(1 - 49*x)^(5/7).

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A224881 Expansion of 1/(1 - 16*x)^(1/8).

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A298799 Expansion of (1-27*x)^(-1/9).

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A386271 Expansion of 1/(1 - 49*x)^(2/7).

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A386272 Expansion of 1/(1 - 49*x)^(3/7).

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A386273 Expansion of 1/(1 - 49*x)^(4/7).

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