A045757 -id:A045757 - cp's OEIS frontend

A035265 One half of deca-factorial numbers.

1, 12, 264, 8448, 354816, 18450432, 1143926784, 82362728448, 6753743732736, 621344423411712, 63377131187994624, 7098238693055397888, 865985120552758542336, 114310035912964127588352, 16232025099640906117545984, 2467267815145417729866989568, 399697386053557672238452310016

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A008548, A045757, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

List([1..20], n-> Product([1..n], j-> 10*j-8)/2 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-8: j in [1..n]])/2: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-8, j=1..n)/2, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[2/10, n]/2, {n,20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-8)/2 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-8) for j in (1..n))/2 for n in (1..20)] # G. C. Greubel, Nov 11 2019

2*a(n) = (10*n-8)(!^10) = Product_{j=1..n} (10*j-8).
a(n) = 2^n*A008548(n) where A008548(n) = (5*n-4)(!^5) = Product_{j=1..n} (5*j-4).
E.g.f.: (-1 + (1-10*x)^(-1/5))/2.
a(n) = (Pochhammer(2/10,n)*10^n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/10^8)^(1/10)*(Gamma(1/5) - Gamma(1/5, 1/10)). - Amiram Eldar, Dec 22 2022

A144773 10-fold factorials: Product_{k=0..n-1} (10*k+1).

Original entry on oeis.org

1, 1, 11, 231, 7161, 293601, 14973651, 913392711, 64850882481, 5252921480961, 478015854767451, 48279601331512551, 5359035747797893161, 648443325483545072481, 84946075638344404495011, 11977396665006561033796551, 1808586896415990716103279201, 291182490322974505292627951361

Offset: 0

Philippe Deléham, Sep 21 2008

G. C. Greubel, Table of n, a(n) for n = 0..320

Essentially a duplicate of A045757.
Cf. k-fold factorials: A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A256268 (combined table).
Cf. A048994, A132393.

R:=PowerSeriesRing(Rationals(), 15); Coefficients(R!(Laplace( (1-10*x)^(-1/10) ))); // G. C. Greubel, Mar 03 2020

G(x):=(1-10*x)^(-1/10): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..14); # Zerinvary Lajos, Apr 03 2009

b = 10; Table[FullSimplify[b^n*Gamma[n + 1/b]/Gamma[1/b]], {n, 0, 14}] (* Michael De Vlieger, Sep 14 2016 *)
Join[{1},FoldList[Times,10 Range[0,15]+1]] (* Harvey P. Dale, Oct 24 2022 *)

Vec(serlaplace( (1-10*x)^(-1/10) +O('x^15) )) \\ G. C. Greubel, Mar 03 2020

[10^n*rising_factorial(1/10,n) for n in (0..15)] # G. C. Greubel, Mar 03 2020

a(n) = Sum_{k = 0..n} (-10)^(n - k) * A048994(n, k).
a(n) = Sum_{k = 0..n} 10^(n - k) * A132393(n, k).
E.g.f.: (1 - 10*x)^(-1/10).
a(n) = A045757(n), n>0.
a(n) = (-9)^n * Sum_{k = 0..n} (10/9)^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
G.f.: 1/Q(0), where Q(k) = 1 - (10*k+1)*x/( 1 - 10*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014
a(n) = 10^n * Gamma(n + 1/10) / Gamma(1/10). - Artur Jasinski Aug 23 2016
a(n) ~ sqrt(2*Pi)*10^n*n^(n-2/5)/(Gamma(1/10)*exp(n)). - Ilya Gutkovskiy, Sep 11 2016
D-finite with recurrence: a(n) - (10*n-9)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1 + (e/10^9)^(1/10)*(Gamma(1/10) - Gamma(1/10, 1/10)). - Amiram Eldar, Dec 22 2022

A048176 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -10, 1, 200, -30, 1, -6000, 1100, -60, 1, 240000, -50000, 3500, -100, 1, -12000000, 2740000, -225000, 8500, -150, 1, 720000000, -176400000, 16240000, -735000, 17500, -210, 1, -50400000000, 13068000000, -1313200000, 67690000, -1960000, 32200, -280, 1, 4032000000000, -1095840000000

Offset: 1

Wolfdieter Lang

a(n,m)= R_n^m(a=0,b=10) in the notation of the given reference.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n) = product(x-10*j,j=0..n-1), n >= 1, E(0,x) := 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
Also the Bell transform of the sequence (-1)^n*A051262(n) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			{1}; {-10,1}; {200,-30,1}; {-6000,1100,-60,1}; ... E(3,x) = 200*x-30*x^2+x^3.

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

First (m=1) (unsigned) column sequence is: A051262(n-1). Row sums (signed triangle): A049212(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A045757(n). b=8: A051187, b=9: A051231.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (-1)^n*n!*10^n, 9); # Peter Luschny, Jan 28 2016

rows = 9;
t = Table[(-1)^n*n!*10^n, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

a(n, m) = a(n-1, m-1) - 10*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n

A020018 Nearest integer to Gamma(n + 1/10)/Gamma(1/10).

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 15, 91, 649, 5253, 47802, 482796, 5359036, 64844333, 849460756, 11977396665, 180858689642, 2911824903230, 49792205845229, 901238925798638, 17213663482753993, 345994636003355265, 7300486819670796088

Offset: 0

Simon Plouffe

nonn

			Gamma(1/10)/Gamma(1/10) = 1, so a(0) = 1.
Gamma(1 + 1/10)/Gamma(1/10) = 1/10 < 1/2, so a(1) = 0.
Gamma(2 + 1/10)/Gamma(1/10) = 11/100 < 1/2, so a(2) = 0.
Gamma(3 + 1/10)/Gamma(1/10) = 231/1000 < 1/2, so a(3) = 0.
Gamma(4 + 1/10)/Gamma(1/10) = 7161/10000 = 0.7161, so a(4) = 1.
Gamma(5 + 1/10)/Gamma(1/10) = 293601/100000 = 2.93601, so a(5) = 3.
Gamma(6 + 1/10)/Gamma(1/10) = 14973651/1000000 = 14.973651, so a(6) = 15.

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A045757, A020063, A020108, A000007 (decimal expansion of 1/10), A256191 (decimal expansion of Gamma(1/10)).

[Round(Gamma(n +1/10)/Gamma(1/10)): n in [0..30]]; // G. C. Greubel, Jan 20 2018

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

Table[Round[Gamma[n + 1/10]/Gamma[1/10]], {n, 0, 50}] (* G. C. Greubel, Jan 20 2018 *)

for(n=0,30, print1(round(gamma(n+1/10)/gamma(1/10)), ", ")) \\ G. C. Greubel, Jan 20 2018

Previous Showing 11-14 of 14 results.

cp's OEIS Frontend

A035265 One half of deca-factorial numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A144773 10-fold factorials: Product_{k=0..n-1} (10*k+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A048176 Generalized Stirling number triangle of first kind.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Mathematica

Formula

A020018 Nearest integer to Gamma(n + 1/10)/Gamma(1/10).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI