A163932 -id:A163932 - cp's OEIS frontend

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

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

A001233 Unsigned Stirling numbers of first kind s(n,6).

Original entry on oeis.org

1, 21, 322, 4536, 63273, 902055, 13339535, 206070150, 3336118786, 56663366760, 1009672107080, 18861567058880, 369012649234384, 7551527592063024, 161429736530118960, 3599979517947607200, 83637381699544802976, 2021687376910682741568, 50779532534302850198976, 1323714091579185857760000

Offset: 6

N. J. A. Sloane

The asymptotic expansion of the higher order exponential integral E(x,m=6,n=1) ~ exp(-x)/x^6*(1 - 21/x + 322/x^2 - 4536/x^3 + 63273/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))^6 = x^6 + 3*x^7 + 23/4*x^8 + 9*x^9 + ...

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).

T. D. Noe, Table of n, a(n) for n = 6..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, A000482, A001234, A008275, A243569, A243570.

Drop[Abs[StirlingS1[Range[30],6]],5] (* Harvey P. Dale, Sep 17 2013 *)

for(n=5,50,print1(polcoeff(prod(i=1,n,x+i),5,x),","))

[stirling_number1(i,6) for i in range(6,22)] # 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^5; or a(n) = P'''''(n-1,0)/5!. - Benoit Cloitre, May 09 2002 [Edited by Petros Hadjicostas, Jun 29 2020 to agree with the offset of 6]
E.g.f.: (-log(1-x))^6/6!.
a(n) is coefficient of x^(n+6) in (-log(1-x))^6, multiplied by (n+6)!/6!.
a(n) = det(|S(i+6,j+5)|, 1 <= i,j <= n-6), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
a(n) = 3*(2*n - 7)*a(n-1) - 5*(3*n^2 - 24*n + 49)*a(n-2) + 10*(2*n - 9)*(n^2 - 9*n + 21)*a(n-3) - (15*n^4 - 300*n^3 + 2265*n^2 - 7650*n + 9751)*a(n-4) + (2*n - 11)*(n^2 - 11*n + 31)*(3*n^2 - 33*n + 91)*a(n-5) - (n-6)^6*a(n-6). - Vaclav Kotesovec, Feb 24 2025

A001234 Unsigned Stirling numbers of the first kind s(n,7).

Original entry on oeis.org

1, 28, 546, 9450, 157773, 2637558, 44990231, 790943153, 14409322928, 272803210680, 5374523477960, 110228466184200, 2353125040549984, 52260903362512720, 1206647803780373360, 28939583397335447760

Offset: 7

N. J. A. Sloane

The asymptotic expansion of the higher order exponential integral E(x,m=7,n=1) ~ exp(-x)/x^7*(1 - 28/x + 546/x^2 - 9450/x^3 + 157773/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

			G.f. = x^7 + 28*x^8 + 546*x^9 + 9450*x^10 + 157773*x^11 + 2637558*x^12 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 834.
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).

T. D. Noe, Table of n, a(n) for n=7..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. A008275 (Stirling1 triangle).

Cf. A000254, A000399, A000454, A000482, A001233, A243569, A243570.

A001234 := proc(n) abs(combinat[stirling1](n,7)) ; end: seq(A001234(n),n=7..30) ; # R. J. Mathar, Nov 06 2009

Table[Abs[StirlingS1[n, 7]], {n, 7, 40}] (* Jean-François Alcover, Mar 24 2020 *)

for(n=6,50,print1(polcoeff(prod(i=1,n,x+i),6,x),","))

[stirling_number1(i,7) for i in range(7,22)] # 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^6; or a(n) = P^(vi)(n-1,0)/6!. - Benoit Cloitre, May 09 2002 [Edited by Petros Hadjicostas, Jun 29 2020 to agree with the offset 7]

a(n) = det(|S(i+7,j+6)|, 1 <= i,j <= n-7), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013

More terms from R. J. Mathar, Nov 06 2009

A163934 Triangle related to the asymptotic expansion of E(x,m=4,n).

Original entry on oeis.org

1, 6, 4, 35, 40, 10, 225, 340, 150, 20, 1624, 2940, 1750, 420, 35, 13132, 27076, 19600, 6440, 980, 56, 118124, 269136, 224490, 90720, 19110, 2016, 84, 1172700, 2894720, 2693250, 1265460, 330750, 48720, 3780, 120

Offset: 1

Johannes W. Meijer, Aug 13 2009

The higher order exponential integrals E(x,m,n) are defined in A163931 while the general formula for their asymptotic expansion can be found in A163932.
We used the latter formula and the asymptotic expansion of E(x,m=3,n), see A163932, to determine that E(x,m=4,n) ~ (exp(-x)/x^4)*(1 - (6+4*n)/x + (35+40*n+ 10*n^2)/x^2 - (225+340*n+ 150*n^2+20*n^3)/x^3 + ... ). This formula leads to the triangle coefficients given above.
The asymptotic expansion leads for the values of n from one to five to known sequences, see the cross-references.
The numerators of the o.g.f.s. of the right hand columns of this triangle lead for z=1 to A000457, see A163939 for more information.
The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=4,n).

			The first few rows of the triangle are:
1;
6, 4;
35, 40, 10;
225, 340, 150, 20;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A163931 (E(x,m,n)), A163932 and A163939.
Cf. A048994 (Stirling1), A000454 (row sums).
A000399, 4*A000454, 10*A000482, 20*A001233, 35*A001234 equal the first five left hand columns.
A000292, A027777 and A163935 equal the first three right hand columns.
The asymptotic expansion leads to A000454 (n=1), A001707 (n=2), A001713 (n=3), A001718 (n=4) and A001723 (n=5).
Cf. A130534 (m=1), A028421 (m=2), A163932 (m=3).

with(combinat): A163934 := proc(n,m): (-1)^(n+m)* binomial(m+2, 3) *stirling1(n+2, m+2) end: seq(seq(A163934(n,m), m=1..n), n=1..8);
with(combinat): imax:=6; EA:=proc(x,m,n) local E, i; E:=0: for i from m-1 to imax+2 do E:=E + sum((-1)^(m+k+1)*binomial(k,m-1)*n^(k-m+1)* stirling1(i, k), k=m-1..i)/x^(i-m+1) od: E:= exp(-x)/x^(m)*E: return(E); end: EA(x,4,n);
# Maple programs revised by Johannes W. Meijer, Sep 11 2012

a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+2, 3] * StirlingS1[n+2, m+2]; Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]] (* Jean-François Alcover, Jun 01 2011, after formula *)

a(n,m) = (-1)^(n+m)*C(m+2,3)*stirling1(n+2,m+2) for n >= 1 and 1<= m <= n.

A001706 Generalized Stirling numbers.

Original entry on oeis.org

1, 9, 71, 580, 5104, 48860, 509004, 5753736, 70290936, 924118272, 13020978816, 195869441664, 3134328981120, 53180752331520, 953884282141440, 18037635241029120, 358689683932346880, 7483713725055744000, 163478034254755584000, 3731670622213083648000

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=3,n=2) ~ exp(-x)/x^3*(1 - 9/x + 71/x^2 - 580/x^3 + 5104/x^4 - 48860/x^5 + the sequence given above). See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009

a(n-1) is equal to -1 times the coefficient of x of the characteristic polynomial of the n X n matrix whose (i,j)-entry is equal to i+3 if i=j and is equal to 1 otherwise. - John M. Campbell, May 24 2011

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.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

Table[-Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1,#2]((((#1+3)))-1)+1&,{n,n}],x],x,1],{n,1,10}] (* John M. Campbell, May 24 2011 *)

E.g.f. (with offset 2): log(1 - x)^2 / (2 * (1 - x)^2).
a(n) = Sum_{k=0..n}(-1)^(n+k)*binomial(k+2, 2)*2^k*stirling1(n+2, k+2). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
a(n-1) = (1/2)*Sum_{i=0..n} binomial(n, i)*A000254(i)*A000254(n-i). - Benoit Cloitre, Mar 09 2004
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n-1) = |f(n,2,2)|, for n>=2. - Milan Janjic, Dec 21 2008
a(n) = (n+3)!*((gamma-1)*Psi(n+4)+2+gamma^2-17*gamma/6+sum(Psi(i+4)/(i+4),i = 0 .. n-1)). - Mark van Hoeij, Oct 26 2011

More terms from Christian G. Bower

A001712 Generalized Stirling numbers.

Original entry on oeis.org

1, 12, 119, 1175, 12154, 133938, 1580508, 19978308, 270074016, 3894932448, 59760168192, 972751628160, 16752851775360, 304473528961920, 5825460745532160, 117070467915075840, 2465958106403712000, 54336917746726272000, 1250216389189281024000

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=3,n=3) ~ exp(-x)/x^3*(1 - 12/x + 119/x^2 - 1175/x^3 + 12154/x^4 - 133938/x^5 + ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009
From Petros Hadjicostas, Jun 11 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) using slightly different notation. They were further examined by Mitrinovic and Mitrinovic (1962).
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_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m. (Because an empty product is by definition 1, we may let 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). (Array A008275 is the same as array A048994 but with no zero row and no zero column.)
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+2}^2(a=-3, 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
Matt Davis, Quadrant Marked Mesh Patterns and the r-Stirling Numbers, arXiv preprint arXiv:1412.0345 [math.CO], 2014.
Matt Davis, Quadrant Marked Mesh Patterns and the r-Stirling Numbers, J. Int. Seq. 18 (2015), #15.10.1.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv preprint arXiv:1201.1323 [math.CO], 2012.
Sergey Kitaev and Jeffrey Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012), #12.4.7.
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 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.
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].
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49. [Annotated scanned copy]

Cf. A001711, A008275, A048994, A163931, A163932.

A001712 := proc(n)
    add((-1)^(n+k)*binomial(k+2, 2)*3^k*Stirling1(n+2, k+2), k=0..n) ;
end proc:
seq(A001712(n), n=0..10) ; # R. J. Mathar, Jun 09 2018

nn = 22; t = Range[0, nn]! CoefficientList[Series[Log[1 - x]^2/(2*(1 - x)^3), {x, 0, nn}], x]; Drop[t, 2] (* T. D. Noe, Aug 09 2012 *)

a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+2, 2)*3^k*stirling(n+2, k+2, 1)) \\ Michel Marcus, Jan 20 2016

b(n) = prod(r=0, n+1, r+3);
c(n) = sum(i=0, n+1, sum(j=i+1, n+1, 1/((3+i)*(3+j))));
for(n=0, 18, print1(b(n)*c(n),",")) \\ Petros Hadjicostas, Jun 11 2020

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+2, 2)*3^k*Stirling1(n+2, k+2). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
E.g.f.: (1 - 7*log(1 - x) + 6*log(1 - x)^2)/(1 - x)^5. - Vladeta Jovovic, Mar 01 2004
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k, i)*Product_{j=0..k-1} (-a-j), then a(n-2) = |f(n,2,3)|, for n >= 2. [Milan Janjic, Dec 21 2008]
Conjecture: a(n) + 3*(-n-3)*a(n-1) + (3*n^2 + 15*n + 19)*a(n-2) - (n+2)^3*a(n-3)=0. - R. J. Mathar, Jun 09 2018
From Petros Hadjicostas, Jun 11 2020: (Start)
a(n) = [x^2] Product_{r=0}^{n+1} (x + 3 + r) = (Product_{r=0}^{n+1} (r+3)) * Sum_{0 <= i < j <= n+1} 1/((3+i)*(3+j)).
Since a(n) = R_{n+2}^2(a=-3, b=-1) and A001711(n) = R_{n+1}^1(a=-3,b=-1), the equation R_{n+2}^2(a=-3,b=-1) = R_{n+1}^1(a=-3,b=-1) + (n+4)*R_{n+1}^2(a=-3,b=-1) implies the following:
(i) a(n) = A001711(n) + (n+4)*a(n-1) for n >= 1.
(ii) a(n) = (n+2)!/2 + (2*n+7)*a(n-1) - (n+3)^2*a(n-2) for n >= 2.
(iii) R. J. Mathar's recurrence above. (End)

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

A001717 Generalized Stirling numbers.

Original entry on oeis.org

1, 15, 179, 2070, 24574, 305956, 4028156, 56231712, 832391136, 13051234944, 216374987520, 3785626465920, 69751622298240, 1350747863435520, 27437426560500480, 583506719443584000, 12969079056388224000, 300749419818102528000, 7265204785551331584000

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=3,n=4) ~ exp(-x)/x^3*(1 - 15/x + 179/x^2 - 2070/x^3 + 24574/x^4 - 305956/x^5 + ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009
From Petros Hadjicostas, Jun 25 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.
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+2}^2(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.

nn = 20; t = Range[0, nn]! CoefficientList[Series[(1 - 9*Log[1 - x] + 10*Log[1 - x]^2)/(1 - x)^6, {x, 0, nn}], x] (* T. D. Noe, Aug 09 2012 *)

a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+2, 2)*4^k*stirling(n+2, k+2, 1)); \\ Michel Marcus, Jan 20 2016

a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(k+2, 2) * 4^k * Stirling1(n+2, k+2). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
E.g.f.: (1 - 9*log(1 - x) + 10*log(1 - x)^2)/(1 - x)^6. - Vladeta Jovovic, Mar 01 2004
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k) * Stirling1(n-k,i) * Product_{j=0..k-1} (-a-j), then a(n-2) = |f(n,2,4)| for n>=2. - Milan Janjic, Dec 21 2008
From Petros Hadjicostas, Jun 26 2020: (Start)
a(n) = [x^2] Product_{r=0..n+1} (x + 4 + r) = (Product_{r=0..n+1} (4 + r)) * Sum_{0 <= i < j <= n+1} 1/((4 + i)*(4 + j)).
Since a(n) = R_{n+2}^2(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) = A001716(n) + (n+5)*a(n-1) for n >= 1;
(ii) a(n) = (n+3)!/6 + (2*n+9)*a(n-1) - (n+4)^2*a(n-2) for n >= 2.
(iii) a(n) = 3*(n+4)*a(n-1) - (3*n^2+21*n+37)*a(n-2) + (n+3)^3*a(n-3) for n >= 3. (End)

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

A001719 Generalized Stirling numbers.

Original entry on oeis.org

1, 30, 625, 11515, 203889, 3602088, 64720340, 1194928020, 22800117076, 450996059800, 9262414989464, 197632289814960, 4381123888865424, 100869322905986496, 2410630110159777216, 59757230054773959552, 1535299458203884231296, 40848249256425236795904

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=5,n=4) ~ exp(-x)/x^5*(1 - 30/x + 625/x^2 - 11515/x^3 + 203889/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

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, M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

Cf. A000254, A001706, A001713.

nn = 24; t = Range[0, nn]! CoefficientList[Series[(Log[1 - x]/(1 - x))^4/24, {x, 0, nn}], x]; Drop[t, 4] (* T. D. Noe, Aug 09 2012 *)

a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+4, 4)*4^k*stirling(n+4, k+4, 1)); \\ Michel Marcus, Jan 20 2016

E.g.f.: (log(1-x)/(1-x))^4/24. - Vladeta Jovovic, May 05 2003
a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+4, 4)*4^k*Stirling1(n+4, k+4). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n-4) = |f(n,4,4)|, for n>=4. - Milan Janjic, Dec 21 2008

More terms from Vladeta Jovovic, May 05 2003

A001709 Generalized Stirling numbers.

Original entry on oeis.org

1, 27, 511, 8624, 140889, 2310945, 38759930, 671189310, 12061579816, 225525484184, 4392554369840, 89142436976320, 1884434077831824, 41471340993035856, 949385215397800224, 22587683825903611680, 557978742043520648256, 14297219701868137003200

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=6,n=2) ~ exp(-x)/x^6*(1 - 27/x + 511/x^2 - 8624/x^3 + 140889/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

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 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.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

nn = 25; t = Range[0, nn]! CoefficientList[Series[-Log[1 - x]^5/(120*(1 - x)^2), {x, 0, nn}], x]; Drop[t, 5] (* T. D. Noe, Aug 09 2012 *)

a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+5, 5)*2^k*stirling(n+5, k+5, 1)); \\ Michel Marcus, Jan 01 2023

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+5, 5)*2^k*Stirling1(n+5, k+5). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
E.g.f.: (6-120*log(1-x)+465*log(1-x)^2-580*log(1-x)^3+261*log(1-x)^4-36*log(1-x)^5)/(6*(1-x)^7). - Vladeta Jovovic, Mar 01 2004
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n-5) = |f(n,5,2)|, for n>=5. [From Milan Janjic, Dec 21 2008]

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

A001708 Generalized Stirling numbers.

Original entry on oeis.org

1, 20, 295, 4025, 54649, 761166, 11028590, 167310220, 2664929476, 44601786944, 784146622896, 14469012689040, 279870212258064, 5667093514231200, 119958395537083104, 2650594302549806976, 61049697873641191296, 1463708634867162093312, 36482312832434713195776

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=5,n=2) ~ exp(-x)/x^5*(1 - 20/x + 295/x^2 - 4025/x^3 + 54649/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

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), 1-77.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

With[{nn=20},Drop[CoefficientList[Series[Log[1-x]^4/(24(1-x)^2),{x,0,nn}], x]Range[0,nn]!,4]] (* Harvey P. Dale, Oct 24 2011 *)

my(x='x+O('x^25)); Vec(serlaplace((log(1-x))^4/(24*(1-x)^2))) \\ Michel Marcus, Feb 04 2022

E.g.f.: ( log ( 1 - x ))^4 / 24 ( 1 - x )^2.
a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+4, 4)*2^k*Stirling1(n+4, k+4). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a - j), then a(n-4) = |f(n,4,2)| for n >= 4. - Milan Janjic, Dec 21 2008

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

cp's OEIS Frontend

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

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A001233 Unsigned Stirling numbers of first kind s(n,6).

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A001234 Unsigned Stirling numbers of the first kind s(n,7).

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A163934 Triangle related to the asymptotic expansion of E(x,m=4,n).

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A001706 Generalized Stirling numbers.

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A001712 Generalized Stirling numbers.

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