A163931 -id:A163931 - cp's OEIS frontend

A049388 a(n) = (n+7)!/7!.

1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200, 4151347200, 70572902400, 1270312243200, 24135932620800, 482718652416000, 10137091700736000, 223016017416192000, 5129368400572416000, 123104841613737984000, 3077621040343449600000, 80018147048929689600000

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher order exponential integral E(x,m=1,n=8) ~ exp(-x)/x*(1 - 8/x + 72/x^2 - 720/x^3 + 7920/x^4 - 95040/x^5 + 235520/x^6 - 17297280/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000142, A001710, A001715, A001720, A001725, A001730, A051339, A051379.

Cf. A130534, A163931, A173333, A245334.

a049388 = (flip div 5040) . a000142 . (+ 7)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+7)/5040: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

((Range[0,20]+7)!)/7! (* Harvey P. Dale, Jul 31 2012 *)

vector(20,n,n--; (n+7)!/7!) \\ G. C. Greubel, Aug 15 2018

a(n)= A051379(n, 0)*(-1)^n (first unsigned column of triangle).
a(n) = (n+7)!/7!.
E.g.f.: 1/(1-x)^8.
a(n) = A173333(n+7,7). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+7,n) / 8. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 5040*e - 13699.
Sum_{n>=0} (-1)^n/a(n) = 1855 - 5040/e. (End)

A195161 Multiples of 8 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 8, 3, 16, 5, 24, 7, 32, 9, 40, 11, 48, 13, 56, 15, 64, 17, 72, 19, 80, 21, 88, 23, 96, 25, 104, 27, 112, 29, 120, 31, 128, 33, 136, 35, 144, 37, 152, 39, 160, 41, 168, 43, 176, 45, 184, 47, 192, 49, 200, 51, 208, 53, 216, 55, 224, 57, 232, 59

Offset: 0

Omar E. Pol, Sep 10 2011

A008590 and A005408 interleaved. This is 8*n if n is even, n if n is odd, if n>=0.
Partial sums give the generalized 12-gonal (or dodecagonal) numbers A195162.
The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (- 4*log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - Johannes W. Meijer, Jul 03 2016
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 12-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Column 8 of A195151.
Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, this sequence, A195312.
Cf. A144433.

&cat[[8*n, 2*n+1]: n in [0..30]]; // Vincenzo Librandi, Sep 27 2011

a := proc(n): (6*(-1)^n+10)*n/4 end: seq(a(n), n=0..59); # Johannes W. Meijer, Jul 03 2016

With[{nn=30},Riffle[8*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,8,3},60] (* Harvey P. Dale, Nov 24 2013 *)

concat(0, Vec(x*(1+8*x+x^2)/((1-x)^2*(1+x)^2) + O(x^99))) \\ Altug Alkan, Jul 04 2016

a(2n) = 8n, a(2n+1) = 2n+1. [corrected by Omar E. Pol, Jul 26 2018]
a(n) = (6*(-1)^n+10)*n/4. - Vincenzo Librandi, Sep 27 2011
a(n) = 2*a(n-2)-a(n-4). G.f.: x*(1+8*x+x^2)/((1-x)^2*(1+x)^2). - Colin Barker, Aug 11 2012
From Ilya Gutkovskiy, Jul 03 2016: (Start)
a(m*2^k) = m*2^(k+2), k>0.
E.g.f.: x*(4*sinh(x) + cosh(x)).
Dirichlet g.f.: 2^(-s)*(2^s + 6)*zeta(s-1). (End)
Multiplicative with a(2^e) = 4*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
a(n) = A144433(n-1) for n > 1. - Georg Fischer, Oct 14 2018

A001716 Generalized Stirling numbers.

Original entry on oeis.org

1, 9, 74, 638, 5944, 60216, 662640, 7893840, 101378880, 1397759040, 20606463360, 323626665600, 5395972377600, 95218662067200, 1773217155225600, 34758188233574400, 715437948072960000, 15429680577561600000, 347968129734973440000, 8190600438533990400000

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=4) ~ exp(-x)/x^2*(1 - 9/x + 74/x^2 - 638/x^3 + 5944/x^4 - 60216/x^5 + 662640/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009
From Petros Hadjicostas, Jun 23 2020: (Start)
For nonnegative integers n, m and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and Mitrinovic and Mitrinovic (1962) using slightly different notation.
These numbers are defined via the g.f. 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_0^0(a,b) = 1, R_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m.
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) for n, m >= 0.
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 sequence, a(n) = R_{n+1}^1(a=-4, b=-1) for n >= 0. (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
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. [The numbers R_n^m(a,b) are introduced.]
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].
D. S. Mitrinovic and M. 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.
J. Riordan, Letter of 04/11/74.

Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564.

f[k_] := k + 3; t[n_] := Table[f[k], {k, 1, n}]; a[n_] := SymmetricPolynomial[n - 1, t[n]]; Table[a[n], {n, 1, 16}] (* Clark Kimberling, Dec 29 2011 *)
Rest[CoefficientList[Series[(1-x)^(-4)*Log[1/(1-x)],{x,0,20}],x]*Range[0,20]!] (* Vaclav Kotesovec, Jan 19 2014 *)

R(n, m, a, b) =  sum(k=0, n-m, (-1)^k*a^k*b^(n-m-k)*binomial(m+k, k)*stirling(n, m+k, 1));
aa(n) = R(n+1, 1, -4, -1);
for(n=0, 19, print1(aa(n), ", ")) \\ Petros Hadjicostas, Jun 23 2020

a(n) = Sum_{k=0..n} (-1)^(n+k) * (k+1) * 4^k * stirling1(n+1, k+1). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
a(n-1) = n!*Sum_{k=0..n-1} (-1)^k*binomial(-4,k)/(n-k) for n >= 1. [Milan Janjic, Dec 14 2008] [Edited by Petros Hadjicostas, Jun 23 2020]
a(n)= n! * [3]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n (with offset 1). [Gary Detlefs, Jan 04 2011]
a(n) = (n+1)! * Sum_{k=0..n} (-1)^k*binomial(-4,k)/(n+1-k). [Gary Detlefs, Jul 16 2011]
a(n) = (n+4)! * Sum_{k=1..n+1} 1/(k+3)/6. [Gary Detlefs, Sep 14 2011]
E.g.f. (for offset 1): 1/(1-x)^4 * log(1/(1-x)). - Vaclav Kotesovec, Jan 19 2014
E.g.f.: (1 + 4*log(1/(1 - x)))/(1 - x)^5. - Ilya Gutkovskiy, Jan 23 2017
From Petros Hadjicostas, Jun 23 2020: (Start)
a(n) = [x] Product_{r=0..n} (x + 4 + r) = (Product_{r=0..n} (4 + r)) * Sum_{i=0..n} 1/(4 + i).
Since a(n) = R_{n+1}^1(a=-4, b=-1) and R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b), we conclude that:
(i) a(n) = (n+3)!/6 + (n+4)*a(n-1) for n >= 1;
(ii) a(n) = (2*n+7)*a(n-1) - (n+3)^2*a(n-2) for n >= 2. (End)

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

A000454 Unsigned Stirling numbers of first kind s(n,4).

Original entry on oeis.org

1, 10, 85, 735, 6769, 67284, 723680, 8409500, 105258076, 1414014888, 20313753096, 310989260400, 5056995703824, 87077748875904, 1583313975727488, 30321254007719424, 610116075740491776

Offset: 4

N. J. A. Sloane

nonn

Number of permutations of n elements with exactly 4 cycles.

The asymptotic expansion of the higher order exponential integral E(x, m=4, n=1) ~ exp(-x)/x^4*(1 - 10/x + 85/x^2 - 735/x^3 + 6769/x^4 - ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Jun 11 2016

			(-log(1-x))^4 = x^4 + 2*x^5 + (17/6)*x^6 + (7/2)*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Shanzhen Gao, Permutations with Restricted Structure (in preparation) [From Shanzhen Gao, Sep 14 2010] [Apparently unpublished as of June 2016]

T. D. Noe, Table of n, a(n) for n=4..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 33.

Cf. A000254, A000399, A000482, A001233, A001234, A008275, A243569, A243570.

Abs[StirlingS1[Range[4,20],4]] (* Harvey P. Dale, Aug 26 2011 *)

for(n=3,50,print1(polcoeff(prod(i=1,n,x+i),3,x),","))

[stirling_number1(i,4) for i in range(4,22)] # Zerinvary Lajos, Jun 27 2008

Let P(n-1,X) = (X+1)(X+2)(X+3)...(X+n-1); then a(n) is the coefficient of X^3; or a(n) = P'''(n-1,0)/3!. - Benoit Cloitre, May 09 2002 [Edited by Petros Hadjicostas, Jun 29 2020 to agree with the offset 4]
E.g.f.: (-log(1-x))^4/4!. [Corrected by Joerg Arndt, Oct 05 2009]
a(n) is coefficient of x^(n+4) in (-log(1-x))^4, multiplied by (n+4)!/4!.
a(n) = (h(n-1, 1)^3 - 3*h(n-1, 1)*h(n-1, 2) + 2*h(n-1, 3))*(n-1)!/3!, where h(n, r) = Sum_{i=1..n} 1/i^r. - Klaus Strassburger, 2000
a(n) = det(|S(i+4,j+3)|, 1 <= i,j <= n-4), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
a(n) = y(n)*n!/24, where y(0) = y(1) = y(2) = y(3) = 0, y(4) = 1 and n^4*y(n) + (-1-5*n-10*n^2-10*n^3-4*n^4)*y(n+1) + (1+n)*(2+n)*(7+12*n+6*n^2)*y(n+2) - 2*(1+n)*(2+n)*(3+n)*(3+2*n)*y(3+n) + (1+n)*(2+n)*(3+n)*(4+n)*y(n+4) = 0. - Benedict W. J. Irwin, Jul 12 2016
From Vaclav Kotesovec, Jul 12 2016: (Start)
a(n) = 2*(2*n - 5)*a(n-1) - (6*n^2 - 36*n + 55)*a(n-2) + (2*n - 7)*(2*n^2 - 14*n + 25)*a(n-3) - (n-4)^4*a(n-4).
a(n) ~ n! * (log(n))^3 / (6*n) * (1 + 3*gamma/log(n) + (3*gamma^2 - Pi^2/2)/ (log(n))^2), where gamma is the Euler-Mascheroni constant A001620. (End)
From Petros Hadjicostas, Jun 29 2020: (Start)
a(n) = A000399(n-1) + (n-1)*a(n-1) for n >= 1 (assuming a(n) = 0 for n = 0..3).
a(n) = A103719(n-4) + (n-2)*a(n-1) for n >= 4.
a(n) = A000254(n-3) + (2*n-3)*a(n-1) - (n-2)^2*a(n-2) for n >= 3.
a(n) = (n-4)! + 3*(n-2)*a(n-1) - (3*n^2-15*n+19)*a(n-2) + (n-3)^3*a(n-3) for n >= 4. (End)

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000

A000482 Unsigned Stirling numbers of first kind s(n,5).

Original entry on oeis.org

1, 15, 175, 1960, 22449, 269325, 3416930, 45995730, 657206836, 9957703756, 159721605680, 2706813345600, 48366009233424, 909299905844112, 17950712280921504, 371384787345228000, 8037811822645051776, 181664979520697076096, 4280722865357147142912, 105005310755917452984576

Offset: 5

N. J. A. Sloane

Number of permutations of n elements with exactly 5 cycles.
Let P(n-1,X) = (X+1)(X+2)(X+3)...(X+n-1); then a(n) is the coefficient of X^4; or a(n) = P''''(n-1,0)/4! - Benoit Cloitre, May 09 2002 [Edited by Petros Hadjicostas, Jun 29 2020 to agree with the offset of 5]
The asymptotic expansion of the higher order exponential integral E(x,m=5,n=1) ~ exp(-x)/x^5*(1 - 15/x + 175/x^2 - 1960/x^3 + 22449/x^4 - ...) leads to the sequence given above. See A163931 for E(x,m,n) information and A163932 for a Maple procedure for the asymptotic expansion. - Johannes W. Meijer, Oct 20 2009

			(-log(1-x))^5 = x^5 + 5/2*x^6 + 25/6*x^7 + 35/6*x^8 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Shanzhen Gao, Permutations with Restricted Structure (in preparation) [Shanzhen Gao, Sep 14 2010]

T. D. Noe, Table of n, a(n) for n=5..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000254, A000399, A000454, A001233, A001234, A008275, A243569, A243570.

Abs[StirlingS1[Range[5,30],5]] (* Harvey P. Dale, May 26 2014 *)

for(n=4,50,print1(polcoeff(prod(i=1,n,x+i),4,x),","))

[stirling_number1(i,5) for i in range(5,22)] # Zerinvary Lajos, Jun 27 2008

E.g.f.: (-log(1-x))^5/5!. [Corrected by Joerg Arndt, Oct 05 2009]
a(n) is coefficient of x^(n+5) in (-log(1-x))^5, multiplied by (n+5)!/5!.
a(n) = det(|S(i+5,j+4)|, 1 <= i,j <= n-5), where S(n,k) are Stirling numbers of the second kind. [Mircea Merca, Apr 06 2013]
a(n) = 5*(n-3)*a(n-1) - 5*(2*n^2 - 14*n + 25)*a(n-2) + 5*(n-4)*(2*n^2 - 16*n + 33)*a(n-3) - (5*n^4 - 90*n^3 + 610*n^2 - 1845*n + 2101)*a(n-4) + (n-5)^5*a(n-5). - Vaclav Kotesovec, Feb 24 2025

A001721 Generalized Stirling numbers.

Original entry on oeis.org

1, 11, 107, 1066, 11274, 127860, 1557660, 20355120, 284574960, 4243508640, 67285058400, 1131047366400, 20099588140800, 376612896038400, 7422410595801600, 153516757766400000, 3325222830101760000, 75283691519393280000, 1778358268603445760000

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=5) ~ exp(-x)/x^2*(1 - 11/x + 107/x^2 - 1066/x^3 + 11274/x^4 - 127860/x^5 + 1557660/x^6 - ... ) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
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, 77 pp.

Related to n!*(the k-th successive summation of the harmonic numbers): k=0..A000254, k=1..A001705,k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564.

f[k_] := k + 4; t[n_] := Table[f[k], {k, 1, n}]; a[n_] := SymmetricPolynomial[n - 1, t[n]]; Table[a[n], {n, 1, 16}] (* Clark Kimberling, Dec 29 2011 *)

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+1, 1)*5^k*Stirling1(n+1, k+1). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
a(n) = n!*Sum_{k=0..n-1} (-1)^k*binomial(-5,k)/(n-k). - Milan Janjic, Dec 14 2008
a(n) = n!*[4]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n. With offset 1. - Gary Detlefs, Jan 04 2011
E.g.f.: (1 + 5*log(1/(1-x)))/(1 - x)^6. - Ilya Gutkovskiy, Jan 23 2017

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

A049389 a(n) = (n+8)!/8!.

Original entry on oeis.org

1, 9, 90, 990, 11880, 154440, 2162160, 32432400, 518918400, 8821612800, 158789030400, 3016991577600, 60339831552000, 1267136462592000, 27877002177024000, 641171050071552000, 15388105201717248000, 384702630042931200000, 10002268381116211200000

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=9) ~ exp(-x)/x*(1 - 9/x + 90/x^2 - 990/x^3 + 11880/x^4 - 154440/x^5 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000142, A001710, A001715, A001720, A001725, A001730, A049388, A051379, A051380.

Cf. A130534, A163931, A173333, A245334.

a049389 = (flip div 40320) . a000142 . (+ 8)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+8)/40320: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_] := (n + 8)!/8!; Array[a, 20, 0] (* Amiram Eldar, Jan 15 2023 *)

PARI
```
a(n) = (n+8)!/8!;
```

a(n)= A051380(n, 0)*(-1)^n (first unsigned column of triangle).
a(n) = (n+8)!/8!.
E.g.f.: 1/(1-x)^9.
a(n) = A173333(n+8,8). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+8,n) / 9. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 40320*e - 109600.
Sum_{n>=0} (-1)^n/a(n) = 40320/e - 14832. (End)

A051524 Second unsigned column of triangle A051338.

Original entry on oeis.org

0, 1, 13, 146, 1650, 19524, 245004, 3272688, 46536624, 703404576, 11277554400, 191338156800, 3427105248000, 64651956364800, 1281740285145600, 26648514872985600, 579892995734169600, 13183403757582643200

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=6) ~ exp(-x)/x^2*(1 - 13/x + 146/x^2 - 1650/x^3 + 19524/x^4 - 245004/x^5 + 3272688/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009

Mitrinovic, D. S. and Mitrinovic, R. S.: see reference given for triangle A051338.

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A001725 (first unsigned column).

Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564. - Gary Detlefs, Jan 04 2011

f[k_] := k + 5; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]
(* Clark Kimberling, Dec 29 2011 *)

a(n) = A051338(n, 1)*(-1)^(n-1);
E.g.f.: -log(1-x)/(1-x)^6.
For n>=1, a(n) = n!*Sum_{k=0..n-1} (-1)^k*binomial(-6,k)/(n-k). - Milan Janjic, Dec 14 2008
a(n) = n!*[5]h(n), where [k]h(n) denotes the k-th successive summation of h(n) from 0 to n. - Gary Detlefs, Jan 04 2011
Conjecture: a(n) +(-2*n-9)*a(n-1) +(n+4)^2*a(n-2)=0. - R. J. Mathar, Aug 04 2013

A051545 Second unsigned column of triangle A051339.

Original entry on oeis.org

0, 1, 15, 191, 2414, 31594, 434568, 6314664, 97053936, 1576890000, 27046454400, 488849155200, 9293295110400, 185464792800000, 3878247384345600, 84822225638169600, 1937048605944883200, 46113230058645657600

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=7) ~ exp(-x)/x^2*(1 - 15/x + 191/x^2 - 2414/x^3 + 31594/x^4 - 434568/x^5 + 6314664/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009

Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051339.

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A001730 (first unsigned column).

Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..(this sequence), k=7..A051560, k=8..A051562, k=9..A051564. - Gary Detlefs, Jan 04 2011

f[k_] := k + 6; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]
(* Clark Kimberling, Dec 29 2011 *)

a(n) = A051339(n, 2)*(-1)^(n-1).
E.g.f.: -log(1-x)/(1-x)^7.
a(n) = n!*Sum_{k=0,..,n-1}((-1)^k*binomial(-7,k)/(n-k)), for n>=1. - Milan Janjic, Dec 14 2008
a(n) = n!*[6]h(n), where [k]h(n) denotes the k-th successive summation of The harmonic numbers from 0 to n. - Gary Detlefs, Jan 04 2011

A051560 Second unsigned column of triangle A051379.

Original entry on oeis.org

0, 1, 17, 242, 3382, 48504, 725592, 11393808, 188204400, 3270729600, 59753750400, 1146140409600, 23046980025600, 485075533132800, 10669304848204800, 244861798361241600, 5854837379724748800

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=8) ~ exp(-x)/x^2*(1 - 17/x + 242/x^2 - 3382/x^3 + 48504/x^4 - 725592/x^5 + 11393808/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009

Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051379.

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A049388 (first unsigned column).

Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564. - Gary Detlefs Jan 04 2011

f[k_] := k + 7; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]
(* Clark Kimberling, Dec 29 2011 *)

a(n) = A051379(n, 2)*(-1)^(n-1).
E.g.f.: -log(1-x)/(1-x)^8.
a(n) = n!*Sum_{k=0..n-1} ((-1)^k*binomial(-8,k)/(n-k)), for n>=1. - Milan Janjic, Dec 14 2008
a(n) = n!*[7]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n. - Gary Detlefs, Jan 04 2011
Conjecture: a(n) +(-2*n-13)*a(n-1) +(n+6)^2*a(n-2)=0. - R. J. Mathar, Aug 04 2013

cp's OEIS Frontend

A049388 a(n) = (n+7)!/7!.

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A195161 Multiples of 8 and odd numbers interleaved.

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A001716 Generalized Stirling numbers.

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A000454 Unsigned Stirling numbers of first kind s(n,4).

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A000482 Unsigned Stirling numbers of first kind s(n,5).

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A001721 Generalized Stirling numbers.

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