A049209 -id:A049209 - cp's OEIS frontend

A147585 a(1) = 1; a(n) = (7n-9)a(n-1) for n > 1.

1, 5, 60, 1140, 29640, 978120, 39124800, 1838865600, 99298742400, 6057223286400, 411891183475200, 30891838760640000, 2533130778372480000, 225448639275150720000, 21643069370414469120000, 2229236145152690319360000, 245215975966795935129600000, 28690269188115124410163200000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..340

Cf. A048994, A132393.

Cf. A045754, A049209, A084947, A144739, A144827, A051188.

[ n eq 1 select 1 else Self(n-1)*(7*n-9): n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008

[ 1 ] cat [ &*[ (5+7*k): k in [0..n-1] ]: n in [1..14] ]; // Klaus Brockhaus, Nov 10 2008

seq( -7^n*pochhammer(-2/7, n)/2, n = 1..15); # G. C. Greubel, Dec 03 2019

Table[-7^n*Pochhammer[-2/7, n]/2, {n, 15}] (* G. C. Greubel, Dec 03 2019 *)

{for(n=1, 15, print1(prod(k=1, n-1, 7*k-2,), ","))} \\ Klaus Brockhaus, Nov 10 2008

[-7^n*rising_factorial(-2/7, n)/2 for n in (1..15)] # G. C. Greubel, Dec 03 2019

a(n) = Product_{k=1..n-1} (7*k - 2). - Klaus Brockhaus, Nov 10 2008
a(n) = (5*7^(n-1)*Gamma(5/7+n))/Gamma(12/7). - Klaus Brockhaus, Nov 10 2008
a(n+1) = Sum_{k=0..n} A132393(n,k)*5^k*7^(n-k). - Philippe Deléham, Nov 09 2008
G.f.: x/(1-5x/(1-7x/(1-12x/(1-14x/(1-19x/(1-21x/(1-26x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-2)^n*Sum_{k=0..n} (7/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
Sum_{n>=1} 1/a(n) = 1 + (e/7^2)^(1/7)*(Gamma(5/7) - Gamma(5/7, 1/7)). - Amiram Eldar, Dec 19 2022

Edited by Klaus Brockhaus, Nov 10 2008

A257627 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(x) = 7*x + 3.

Original entry on oeis.org

1, 3, 3, 9, 60, 9, 27, 753, 753, 27, 81, 8178, 25602, 8178, 81, 243, 84291, 631506, 631506, 84291, 243, 729, 852144, 13348623, 30312288, 13348623, 852144, 729, 2187, 8554245, 259308063, 1141302225, 1141302225, 259308063, 8554245, 2187

Offset: 0

Dale Gerdemann, May 10 2015

			Array t(n, k) begins as:
    1,       3,          9,            27,              81, ... A000244;
    3,      60,        753,          8178,           84291, ...;
    9,     753,      25602,        631506,        13348623, ...;
   27,    8178,     631506,      30312288,      1141302225, ...;
   81,   84291,   13348623,    1141302225,     70760737950, ...;
  243,  852144,  259308063,   37244959794,   3608891348622, ...;
  729, 8554245, 4793178096, 1109572049376, 161806374029202, ...;
Triangle, T(n, k) begins as:
     1;
     3,       3;
     9,      60,         9;
    27,     753,       753,         27;
    81,    8178,     25602,       8178,         81;
   243,   84291,    631506,     631506,      84291,       243;
   729,  852144,  13348623,   30312288,   13348623,    852144,     729;
  2187, 8554245, 259308063, 1141302225, 1141302225, 259308063, 8554245, 2187;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000244, A038221, A049209 (row sums), A142462.
Cf. A257180, A257611, A257617, A257620, A257621, A257623, A257625.
See similar sequences listed in A256890.

f[n_]:= 7*n+3;
t[n_, k_]:= t[n,k]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, f[k]*t[n-1,k] +f[n]*t[n,k-1]]];
T[n_, k_]= t[n-k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)

def f(n): return 7*n+3
@CachedFunction
def t(n,k):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return f(k)*t(n-1, k) + f(n)*t(n, k-1)
def A257627(n,k): return t(n-k,k)
flatten([[A257627(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2022

T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 3.
Sum_{k=0..n} T(n, k) = A049209(n).
From G. C. Greubel, Feb 22 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A254322 Expansion of e.g.f.: (1-11*x)^(-10/11).

Original entry on oeis.org

1, 10, 210, 6720, 288960, 15603840, 1014249600, 77082969600, 6706218355200, 657209398809600, 71635824470246400, 8596298936429568000, 1126115160672273408000, 159908352815462823936000, 24465977980765812062208000, 4012420388845593178202112000

Offset: 0

Vaclav Kotesovec, Jan 28 2015

Generally, for k > 1, if e.g.f. = (1-k*x)^(-(k-1)/k) then a(n) ~ n! * k^n / (n^(1/k) * Gamma((k-1)/k)).

G. C. Greubel, Table of n, a(n) for n = 0..300

Sequences of the form k^n*Pochhammer((k-1)/k, n): A000007 (k=1), A001147 (k=2), A008544 (k=3), A008545 (k=4), A008546 (k=5), A008543 (k=6), A049209 (k=7), A049210 (k=8), A049211 (k=9), A049212 (k=10), this sequence (k=11), A346896 (k=12).

Cf. A000108, A254282, A254286, A254287.

m=11; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022

CoefficientList[Series[(1-11*x)^(-10/11), {x, 0, 20}], x] * Range[0, 20]!
FullSimplify[Table[11^n * Gamma[n+10/11] / Gamma[10/11], {n, 0, 18}]]

m=11; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

D-finite with recurrence: a(0) = 1; a(n) = (11*n-1) * a(n-1) for n > 0. [corrected by Georg Fischer, Dec 23 2019]
a(n) = 11^n * Gamma(n+10/11) / Gamma(10/11).
a(n) ~ n! * 11^n / (n^(1/11) * Gamma(10/11)).
From Nikolaos Pantelidis, Jan 17 2021: (Start)
G.f.: 1/G(0) where G(k) = 1 - (22*k+10)*x - 11*(k+1)*(11*k+10)*x^2/G(k+1) (continued fraction).
G.f.: 1/(1-10*x-110*x^2/(1-32*x-462*x^2/(1-54*x-1056*x^2/(1-76*x-1892*x^2/(1-98*x-2970*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(11*k+10)/(1 - x*(11*k+11)/Q(k+1)) (continued fraction).
G.f.: 1/(1-10*x/(1-11*x/(1-21*x/(1-22*x/(1-32*x/(1-33*x/(1-43*x/(1-44*x/(1-54*x/(1-55*x/(1-...))))))))))) (Stieltjes continued fraction).
(End)
G.f.: hypergeometric2F0([1, 10/11], [--], 11*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/11)^(1/11)*(Gamma(10/11) - Gamma(10/11, 1/11)). - Amiram Eldar, Dec 22 2022

A034831 a(n) = n-th sept-factorial number divided by 4.

Original entry on oeis.org

1, 11, 198, 4950, 158400, 6177600, 284169600, 15060988800, 903659328000, 60545174976000, 4480342948224000, 362907778806144000, 31935884534940672000, 3033909030819363840000, 309458721143575111680000, 33731000604649687173120000, 3912796070139363712081920000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..339
Index entries for sequences related to factorial numbers.

Cf. A049209, A045754, A034829, A034830, A034832, A034833, A034834, A144827.

[(&*[(7*k-3): k in [1..n]])/4: n in [1..30]]; // G. C. Greubel, Feb 24 2018

Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 7*x)^(-4/7))/4, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)
Table[Product[7 j - 3, {j, n}], {n, 30}]/4 (* Vincenzo Librandi, Feb 24 2018 *)

my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-4/7))/4)) \\ G. C. Greubel, Feb 22 2018

4*a(n) = (7*n-3)(!^7) = Product_{j=1..n} (7*j-3).
E.g.f.: (-1 + (1-7*x)^(-4/7))/4.
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A144827(n)/4.
Sum_{n>=1} 1/a(n) = 4*(e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). (End)

More terms added by G. C. Greubel, Feb 23 2018

A051186 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -7, 1, 98, -21, 1, -2058, 539, -42, 1, 57624, -17150, 1715, -70, 1, -2016840, 657874, -77175, 4165, -105, 1, 84707280, -29647548, 3899224, -252105, 8575, -147, 1, -4150656720, 1537437132, -220709524, 16252369, -672280, 15778, -196, 1

Offset: 1

Wolfdieter Lang

T(n,m) = R_n^m(a=0, b=7) in the notation of the given 1962 reference.
T(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} T(n,m)*x^m = Product_{j=0..n-1} (x-7*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
From Petros Hadjicostas, Jun 07 2020: (Start)
For integers n, m >= 0 and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and further examined by Mitrinovic and Mitrinovic (1962).
They are defined via Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0. As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, R_n^m(a,b) = 0 for n < m, and R_0^0(a,b) = 1.
With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m).
We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.
For the current array, T(n,m) = R_n^m(a=0, b=7) but with no zero row or column. (End)

			Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
         1;
        -7,      1;
        98,    -21,      1;
     -2058,    539,    -42,    1;
     57624, -17150,   1715,  -70,    1;
  -2016840, 657874, -77175, 4165, -105, 1;
  ...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x - 7*j) = 98*x - 21*x^2 + x^3.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Wolfdieter Lang, First ten rows.
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356.
D. S. Mitrinovic and R. S. Mitrinovic, Sur les nombres de Stirling et les nombres de Bernoulli d'ordre supérieur, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 43 (1960), 1-63.
D. S. Mitrinovic and R. S. Mitrinovic, Sur une classe de nombres se rattachant aux nombres de Stirling--Appendice: Table des nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 60 (1961), 1-15 and 17-62.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].

Cf. A000142, A045754 (unsigned row sums), A049209 (row sums), A051188.

The b=1..6 triangles are: A008275 (Stirling1 triangle), A039683, A051141, A051142, A051150, A051151.

[7^(n-k)*StirlingFirst(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 22 2022

Table[7^(n-k)*StirlingS1[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)

flatten([[(-7)^(n-k)*stirling_number1(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 22 2022

T(n, m) = T(n-1, m-1) - 7*(n-1)*T(n-1, m) for n >= m >= 1, T(n, m) = 0 for n < m, T(n, 0) = 0 for n >= 1, and T(0, 0) = 1.
T(n, 1) = A051188(n-1).
Sum_{k=0..n} T(n, k) = (-1)^(n-1)*A049209(n-1).
Sum_{k=0..n} (-1)^(n-k)*T(n, k) = A045754(n).
E.g.f. for m-th column of signed triangle: (log(1 + 7*x)/7)^m/m!.
T(n,m) = 7^(n-m)*S1(n,m) with the (signed) Stirling1 triangle S1(n,m) = A008275(n,m).
Bivariate e.g.f.-o.g.f.: Sum_{n,m >= 1} T(n,m)*x^n*y^m/n! = exp((y/7)*log(1 + 7*x)) - 1 = (1 + 7*x)^(y/7) - 1. - Petros Hadjicostas, Jun 07 2020
T(n, 0) = (-7)^(n-1)*A000142(n-1). - G. C. Greubel, Feb 22 2022

A346896 Expansion of e.g.f.: (1-12*x)^(-11/12).

Original entry on oeis.org

1, 11, 253, 8855, 416185, 24554915, 1743398965, 144702114095, 13746700839025, 1470896989775675, 175036741783305325, 22929813173612997575, 3278963283826658653225, 508239308993132091249875, 84875964601853059238729125, 15192797663731697603732513375

Offset: 0

Nikolaos Pantelidis, Aug 06 2021

G. C. Greubel, Table of n, a(n) for n = 0..300

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), this sequence (m=12).

m:=12; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 16 2022

CoefficientList[Series[(1-12*x)^(-11/12),{x,0,20}], x] * Range[0, 20]!
FullSimplify[Table[12^n Gamma[n+11/12]/Gamma[11/12],{n,0,15}]] (* Stefano Spezia, Aug 07 2021 *)

m=12; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 16 2022

G.f.: 1/(1-11*x/(1-12*x/(1-23*x/(1-24*x/(1-35*x/(1-36*x/(1-47*x/(1-48*x/(1-59*x/(1-60*x/(1-...))))))))))) (Stieltjes continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(12*k+11)/(1 - x*(12*k+12)/Q(k+1) ) (continued fraction).
G.f.: 1/(1-11*x-132*x^2/(1-35*x-552*x^2/(1-59*x-1260*x^2/(1-83*x-2256*x^2/(1-107*x-3540*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/G(0) where G(k) = 1 - x*(24*k+11) - 12*(k+1)*(12*k+11)*x^2/G(k+1) (continued fraction).
a(n) = 12^n*Gamma(n+11/12)/Gamma(11/12). - Stefano Spezia, Aug 07 2021
Sum_{n>=0} 1/a(n) = 1 + (e/12)^(1/12)*(Gamma(11/12) - Gamma(11/12, 1/12)). - Amiram Eldar, Dec 22 2022

A088996 Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 7, 6, 0, 6, 29, 46, 24, 0, 24, 146, 329, 326, 120, 0, 120, 874, 2521, 3604, 2556, 720, 0, 720, 6084, 21244, 39271, 40564, 22212, 5040, 0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320

Offset: 0

Philippe Deléham, Dec 01 2003, Aug 17 2007

			Triangle begins:
  1;
  0,    1;
  0,    1,     2;
  0,    2,     7,      6;
  0,    6,    29,     46,     24;
  0,   24,   146,    329,    326,    120;
  0,  120,   874,   2521,   3604,   2556,    720;
  0,  720,  6084,  21244,  39271,  40564,  22212,   5040;
  0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Trevor Hyde, Liminal reciprocity and factorization statistics, arXiv:1803.08438 [math.NT], 2018.

Variant: A059364, diagonals give A000007, A000142, A067318.

Cf. A001147 (row sums), A048994, A084938.

A088996:= func< n,k | (&+[(-1)^j*Binomial(j,n-k)*StirlingFirst(n,n-j): j in [0..n]]) >;
[A088996(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 23 2022

A059364 := (n, k) -> add(abs(Stirling1(n, n - j))*binomial(j, n - k), j = 0..n);
seq(seq(A059364(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Aug 27 2025

T[n_, k_]:= T[n, k]= Sum[(-1)^(n-i)*Binomial[i, k] StirlingS1[n+1, n+1-i], {i, 0, n}]; {{1}}~Join~Table[Abs@ T[n, k], {n,0,10}, {k,n+1,0,-1}] (* Michael De Vlieger, Jun 19 2018 *)

def A088996(n,k): return add((-1)^(n-i)*binomial(i,k)*stirling_number1(n+1,n+1-i) for i in (0..n))
for n in (0..10): [A088996(n,k) for k in (0..n)]  # Peter Luschny, May 12 2013

T(n, k) given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938. [Original name.]
Sum_{k=0..n} (-1)^k*T(n,k) = (-1)^n.
From Vladeta Jovovic, Dec 15 2004: (Start)
E.g.f.: (1-y-y*x)^(-1/(1+x)).
Sum_{k=0..n} T(n, k)*x^k = Product_{k=1..n} (k*x+k-1). (End)
T(n, k) = n*T(n-1, k-1) + (n-1)*T(n-1, k); T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0. - Philippe Deléham, May 22 2005
Sum_{k=0..n} T(n,k)*x^(n-k) = A019590(n+1), A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, respectively. Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000007(n), A001147(n), A008544(n), A008545(n), A008546(n), A008543(n), A049209(n), A049210(n), A049211(n), A049212(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Aug 10 2007
T(n, k) = Sum_{j=0..n} (-1)^j*binomial(j, n-k)*StirlingS1(n, n-j). - G. C. Greubel, Feb 23 2022

New name using a formula of G. C. Greubel by Peter Luschny, Aug 27 2025

A349971 Array read by ascending antidiagonals, A(n, k) = -(-n)^k*FallingFactorial(1/n, k) for n, k >= 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 15, 0, 1, 4, 21, 80, 105, 0, 1, 5, 36, 231, 880, 945, 0, 1, 6, 55, 504, 3465, 12320, 10395, 0, 1, 7, 78, 935, 9576, 65835, 209440, 135135, 0, 1, 8, 105, 1560, 21505, 229824, 1514205, 4188800, 2027025, 0

Offset: 1

Peter Luschny, Dec 21 2021

			Array starts:
[1] 1, 0,   0,    0,      0,       0,         0,           0, ... A000007
[2] 1, 1,   3,   15,    105,     945,     10395,      135135, ... A001147
[3] 1, 2,  10,   80,    880,   12320,    209440,     4188800, ... A008544
[4] 1, 3,  21,  231,   3465,   65835,   1514205,    40883535, ... A008545
[5] 1, 4,  36,  504,   9576,  229824,   6664896,   226606464, ... A008546
[6] 1, 5,  55,  935,  21505,  623645,  21827575,   894930575, ... A008543
[7] 1, 6,  78, 1560,  42120, 1432080,  58715280,  2818333440, ... A049209
[8] 1, 7, 105, 2415,  74865, 2919735, 137227545,  7547514975, ... A049210
[9] 1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, ... A049211
Triangle starts:
[1] [1]
[2] [1, 0]
[3] [1, 1,  0]
[4] [1, 2,  3,   0]
[5] [1, 3, 10,  15,    0]
[6] [1, 4, 21,  80,  105,     0]
[7] [1, 5, 36, 231,  880,   945,      0]
[8] [1, 6, 55, 504, 3465, 12320,  10395,      0]
[9] [1, 7, 78, 935, 9576, 65835, 209440, 135135, 0]

G. C. Greubel, Antidiagonals n = 1..50, flattened

Rows 1-9: A000007, A001147, A008544, A008545, A008546, A008543, A049209, A049210, A049211.
Main diagonal A349731.
Cf. A158886, A256268.

[k eq n select 0^(n-1) else Round((n-k+1)^(k-1)*Gamma(k-1 + (n-k)/(n-k+1))/Gamma((n-k)/(n-k+1))): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 22 2022

A[n_, k_] := -(-n)^k * FactorialPower[1/n, k]; Table[A[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 21 2021 *)

def A(n, k): return -(-n)^k*falling_factorial(1/n, k)
def T(n, k): return A(n-k+1, k)
for n in (1..9): print([A(n, k) for k in (1..8)])
for n in (1..9): print([T(n, k) for k in (1..n)])

From G. C. Greubel, Feb 22 2022: (Start)
A(n, k) = n^(k-1)*Pochhammer((n-1)/n, k-1) (array).
T(n, k) = (n-k+1)^(k-1)*Pochhammer((n-k)/(n-k+1), k-1) (antidiagonal triangle).
T(2*n, n) = (-1)^(n-1)*A158886(n). (End)

A020029 Nearest integer to Gamma(n + 6/7)/Gamma(6/7).

Original entry on oeis.org

1, 1, 2, 5, 18, 85, 499, 3422, 26889, 238158, 2347553, 25487720, 302211543, 3885576978, 53842995268, 799953072548, 12684970150411, 213832353964067, 3818434892215474, 72004772253206073, 1429809049027949160

Offset: 0

Simon Plouffe

nonn

Gamma(n + 6/7)/Gamma(6/7) = 1, 6/7, 78/49, 1560/343, 42120/2401, 1432080/16807, 58715280/117649, ... - R. J. Mathar, Sep 04 2016

Cf. A049209, A020074, A020119, A220607.

Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;

Table[Round[Gamma[n+6/7]/Gamma[6/7]],{n,0,20}] (* Harvey P. Dale, Dec 16 2023 *)

cp's OEIS Frontend

A147585 a(1) = 1; a(n) = (7*n-9)*a(n-1) for n > 1.

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A257627 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.

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A254322 Expansion of e.g.f.: (1-11*x)^(-10/11).

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A034831 a(n) = n-th sept-factorial number divided by 4.

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A051186 Generalized Stirling number triangle of first kind.

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A346896 Expansion of e.g.f.: (1-12*x)^(-11/12).

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A088996 Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.

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A147585 a(1) = 1; a(n) = (7n-9)a(n-1) for n > 1.

A257627 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(x) = 7*x + 3.