A001712 -id:A001712 - cp's OEIS frontend

A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

1, 3, 3, 11, 18, 6, 50, 105, 60, 10, 274, 675, 510, 150, 15, 1764, 4872, 4410, 1750, 315, 21, 13068, 39396, 40614, 19600, 4830, 588, 28, 109584, 354372, 403704, 224490, 68040, 11466, 1008, 36, 1026576, 3518100, 4342080, 2693250, 949095, 198450

Offset: 1

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Oct 22 2009

The higher order exponential integrals E(x,m,n) are defined in A163931. The general formula for the asymptotic expansion E(x,m,n) ~ E(x,m-1,n+1)/x - n*E(x,m-1,n+2)/x^2 + n*(n+1) * E(x,m-1,n+3)/x^3 - n*(n+1)*(n+2)*E(x,m-1,n+4)/x^4 + ...., m >= 1 and n >= 1.
We used this formula and the asymptotic expansion of E(x,m=2,n), see A028421, to determine that E (x,m=3,n) ~ (exp(-x)/x^3)*(1 - (3+3*n)/x + (11+18*n+6*n^2)/x^2 - (50+105*n+ 60*n^2+ 10*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 ten 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 A001879, see A163938 for more information.
The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=3,n).

			The first few rows of the triangle are:
[1]
[3, 3]
[11, 18, 6]
[50, 105, 60, 10]

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

Cf. A163931 (E(x,m,n)) and A163938.
Cf. A048994 (Stirling1), A000399 (row sums).
A000254, 3*A000399, 6*A000454, 10*A000482, 15*A001233, 21*A001234 equal the first six left hand columns.
A000217, A006011 and A163933 equal the first three right hand columns.
The asymptotic expansion leads to A000399 (n=1), A001706 (n=2), A001712 (n=3), A001717 (n=4), A001722 (n=5), A051525 (n=6), A051546 (n=7), A051561 (n=8), A051563 (n=9) and A051565 (n=10).
Cf. A130534 (m=1), A028421 (m=2) and A163934 (m=4).

nmax:=8; with(combinat): for n1 from 1 to nmax do for m from 1 to n1 do a(n1, m) := (-1)^(n1+m)*binomial(m+1, 2)*stirling1(n1+1, m+1) od: od: seq(seq(a(n1,m), m=1..n1), n1=1..nmax);
# End program 1
with(combinat): imax:=6; EA:=proc(x, m, n) local E, i; E := 0: for i from m-1 to imax+1 do E := E + sum((-1)^(m+k1+1)*binomial(k1, m-1)*n^(k1-m+1)* stirling1(i, k1), k1=m-1..i)/x^(i-m+1) od: E := exp(-x)/x^(m)*E: return(E); end: EA(x, 3, n);
# End program 2

a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+1, 2] * StirlingS1[n+1, m+1]; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 42]] (* Jean-François Alcover, Jun 01 2011, after formula *)

for(n=1,10, for(m=1,n, print1((-1)^(n+m)*binomial(m+1,2) *stirling(n+1,m+1,1), ", "))) \\ G. C. Greubel, Aug 08 2017

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

Edited by Johannes W. Meijer, Sep 22 2012

A001713 Generalized Stirling numbers.

Original entry on oeis.org

1, 18, 245, 3135, 40369, 537628, 7494416, 109911300, 1698920916, 27679825272, 474957547272, 8572072384512, 162478082312064, 3229079010579072, 67177961946534528, 1460629706845766400, 33139181950164806400, 783398920650352012800, 19268391564147377318400

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=4,n=3) ~ exp(-x)/x^4*(1 - 18/x + 245/x^2 - 3135/x^3 + 40369/x^4 - 537628/x^5 + ...) leads to the sequence given above. See A163931 and A163934 for more information. - Johannes W. Meijer, Oct 20 2009
From Petros Hadjicostas, Jun 12 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+3}^3(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
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.

Cf. A000254, A001706, A001711, A001712, A001719, A048994.

nn = 23; t = Range[0, nn]! CoefficientList[Series[-Log[1 - x]^3/(6*(1 - x)^3), {x, 0, nn}], x]; Drop[t, 3] (* T. D. Noe, Aug 09 2012 *)

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

b(n) = prod(r=0, n+2, r+3);
c(n) = sum(i=0, n+2, sum(j=i+1, n+2, sum(k=j+1, n+2, 1/((3+i)*(3+j)*(3+k)))));
for(n=0, 18, print1(b(n)*c(n), ", ")) \\ Petros Hadjicostas, Jun 12 2020

E.g.f.: Sum_{n>=0} a(n)*x^(n+3)/(n+3)! = (log(1 - x)/(x - 1))^3/6. - Vladeta Jovovic, May 05 2003 [Edited by Petros Hadjicostas, Jun 13 2020]
a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(k+3, 3) * 3^k * Stirling1(n+3, k+3). - 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-3) = |f(n,3,3)| for n >= 3. - Milan Janjic, Dec 21 2008
From Petros Hadjicostas, Jun 12 2020: (Start)
a(n) = [x^3] Product_{r=0}^{n+2} (x + 3 + r) = (Product_{r=0}^{n+2} (r+3)) * Sum_{0 <= i < j < k <= n+2} 1/((3+i)*(3+j)*(3+k)).
Since a(n) = R_{n+3}^3(a=-3, b=-1), A001712(n) = R_{n+2}^2(a=-3,b=-1), and A001711(n) = R_{n+1}^1(a=-3, b=-1), the equation R_{n+3}^3(a=-3,b=-1) = R_{n+2}^2(a=-3,b=-1) + (n+5)*R_{n+2}^3(a=-3,b=-1) implies the following:
(i) a(n) = A001712(n) + (n+5)*a(n-1) for n >= 1.
(ii) a(n) = A001711(n) + (2*n+9)*a(n-1) - (n+4)^2*a(n-2) for n >= 2.
(iii) a(n) = (n+2)!/2 + 3*(n+4)*a(n-1) - (3*n^2+21*n+37)*a(n-2) + (n+3)^3*a(n-3) for n >= 3.
(iv) a(n) = 2*(2*n+7)*a(n-1) - (6*n^2+36*n+55)*a(n-2) + (2*n^2+10*n+13)*(2*n+5)*a(n-3) - (n+2)^4*a(n-4) for n >= 4. (End)

More terms from Vladeta Jovovic, May 05 2003

A001714 Generalized Stirling numbers.

Original entry on oeis.org

1, 25, 445, 7140, 111769, 1767087, 28699460, 483004280, 8460980836, 154594537812, 2948470152264, 58696064973000, 1219007251826064, 26390216795274288, 594982297852020288, 13955257961738192448, 340154857108405040256, 8606960634143667938688

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher-order exponential integral E(x,m=5,n=3) ~ exp(-x)/x^5*(1 - 25/x + 445/x^2 - 7140/x^3 + 111769/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
From Petros Hadjicostas, Jun 13 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+4}^4(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
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.

Cf. A001711, A001712, A001713, A048994, A163931, A163932.

nn = 24; t = Range[0, nn]! CoefficientList[Series[Log[1 - x]^4/(24*(1 - x)^3), {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) * 3^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,4,3)| for n >= 4. - Milan Janjic, Dec 21 2008
From Petros Hadjicostas, Jun 14 2020: (Start)
a(n) = [x^4] Product_{r=0}^{n+3} (x + 3 + r) = (Product_{r=0}^{n+3} (r+3)) * Sum_{0 <= i < j < k < m <= n+3} 1/((3+i)*(3+j)*(3+k)*(3+m)).
E.g.f.: Sum_{n>=0} a(n)*x^(n+4)/(n+4)! = (log(1 - x))^4/(1 - x)^3/24.
Since a(n) = R_{n+4}^4(a=-3, b=-1), A001713(n) = R_{n+3}^3(a=-3,b=-1), A001712(n) = R_{n+2}^2(a=-3, b=-1), and A001711(n) = R_{n+1}^1(a=-3,b=-1), the equation R_{n+4}^4(a=-3,b=-1) = R_{n+3}^3(a=-3,b=-1) + (n+6)*R_{n+3}^4(a=-3,b=-1) implies the following:
(i) a(n) = A001713(n) + (n+6)*a(n-1) for n >= 1.
(ii) a(n) = A001712(n) + (2*n+11)*a(n-1) - (n+5)^2*a(n-2) for n >= 2.
(iii) a(n) = A001711(n) + 3*(n+5)*a(n-1) - (3*n^2+27*n+61)*a(n-2) + (n+4)^3*a(n-3) for n >= 3.
(iv) a(n) = (n+2)!/2 + 2*(2*n+9)*a(n-1) - (6*n^2+48*n+97)*a(n-2) + (2*n+7)*(2*n^2+14*n+25)*a(n-3) - (n+3)^4*a(n-4) for n >= 4.
(v) By taking the difference a(n) - (n+2)*a(n-1), and using (iv) above, we get a 5th-order linear recurrence with polynomial coefficients of degree at most 5. We omit the details. (End)

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

A196845 Table of elementary symmetric function a_k(3,4,...,n+2) (no 1 and 2).

Original entry on oeis.org

1, 1, 3, 1, 7, 12, 1, 12, 47, 60, 1, 18, 119, 342, 360, 1, 25, 245, 1175, 2754, 2520, 1, 33, 445, 3135, 12154, 24552, 20160, 1, 42, 742, 7140, 40369, 133938, 241128, 181440, 1, 52, 1162, 14560, 111769, 537628, 1580508, 2592720, 1814400, 1, 63, 1734, 27342, 271929, 1767087, 7494416, 19978308, 30334320, 19958400

Offset: 0

Wolfdieter Lang, Oct 26 2011

For the symmetric functions a_k see a comment in A196841.
In general the triangle S_{i,j}(n,k), n>=k>=0, 1<=i=i as a_k(1,2,...,i-1,i+1,...,j-1,j+1,...,n+2).
  a_0():=1. The present triangle is S_{1,2}(n,k) (no 1 and 2 admitted).

			n\k  0   1    2     3     4       5       6       7  ...
0:   1
1:   1   3
2:   1   7   12
3:   1  12   47    60
4:   1  18  119   342   360
5:   1  25  245  1175  2754    2520
6:   1  33  445  3135 12154   24552   20160
7:   1  42  742  7140 40369  133938  241128  181440
...
a(3,2) = a_2(3,4,5) = 3*4+3*5+4*5 = 47.
a(3,2) = 1*(|s(6,4)| - (1*14 + 2*13)) + 2*(|s(6,6)| -(1*0+2*0)) = 85 - 40 + 2(1-0) = 47.
a(4,3) =  a_3(3,4,5,6) = 3*4*5+3*4*6+3*5*6+4*5*6 = 342.
a(4,3) = 1*(|s(7,4)| - (1*155 + 2*137)) + 2*(|s(7,6)| - (1*1 + 2*1)) = 735-429+2*(21-3) = 342.

Cf. A196841, A048994, A145324, A001710 (diagonal), A001711 (1st subdiagonal), A001712 (2nd subdiagonal), A055998 (k=1), A024183 (k=2), A024184 (k=3), A024185 (k=4).

a(n,k) = 0 if n=0, k=0,...,n, with the elementary symmetric function a_k (see the comment above).

a(n,k) = sum(2^k*( |s(n+3,n+3-k+2*p)| -(S_1(n+1,k-1-2*p) +2*S_2(n+1,k-1-2*p))), p=0..floor(k/2)), with the Stirling numbers of the first kind s(n,m) = A048994(n,m), and the number triangles S_1(n,k)= A145324(n+1,k+1) and S_2(n,k) = A196841(n,k).

A325137 Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 5, 1, 6, 26, 12, 1, 24, 154, 119, 22, 1, 120, 1044, 1175, 355, 35, 1, 720, 8028, 12154, 5265, 835, 51, 1, 5040, 69264, 133938, 77224, 17360, 1687, 70, 1, 40320, 663696, 1580508, 1155420, 342769, 46816, 3066, 92, 1

Offset: 0

Peter Luschny, Apr 13 2019

Sister triangle of A307419.

			Triangle starts:
[0]      1
[1]      1,       1
[2]      2,       5,        1
[3]      6,      26,       12,        1
[4]     24,     154,      119,       22,       1
[5]    120,    1044,     1175,      355,      35,       1
[6]    720,    8028,    12154,     5265,     835,      51,      1
[7]   5040,   69264,   133938,    77224,   17360,    1687,     70,    1
[8]  40320,  663696,  1580508,  1155420,  342769,   46816,   3066,   92,   1
[9] 362880, 6999840, 19978308, 17893196, 6687009, 1197273, 109494, 5154, 117, 1
   A000142, A001705,  A001712,  A001718, A001724, ...

Row sums: A325138.
Columns are: A000142, A001705, A001712, A001718, A001724.
Cf. A307419.

T := (n, k) -> add(binomial(j+k, k)*(k+1)^j*abs(Stirling1(n, j+k)), j=0..n-k);
seq(seq(T(n,k), k=0..n), n=0..8);
# Note that for n > 16 Maple fails (at least in some versions) to compute the
# terms properly. Inserting 'simplify' or numerical evaluation might help.
A325137Row := proc(n) local ogf, ser; ogf := (n, k) -> (n+k+x)!/(k+x)!;
ser := (n, k) -> series(ogf(n,k),x,k+2); seq(coeff(ser(n,k),x,k), k=0..n) end: seq(A325137Row(n), n=0..8);

T(n, k) = Sum_{j=0..n-k} binomial(j+k, k)*|Stirling1(n, j+k)|*(k+1)^j.

A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)binomial(n+m+1, n-k-i)(n + m - k)!/(i + m)!, for m = 2.

Original entry on oeis.org

1, 9, 1, 71, 12, 1, 580, 119, 15, 1, 5104, 1175, 179, 18, 1, 48860, 12154, 2070, 251, 21, 1, 509004, 133938, 24574, 3325, 335, 24, 1, 5753736, 1580508, 305956, 44524, 5000, 431, 27, 1, 70290936, 19978308, 4028156, 617624, 74524, 7155, 539, 30, 1, 924118272, 270074016, 56231712, 8969148, 1139292, 117454, 9850, 659, 33, 1, 13020978816, 3894932448, 832391136, 136954044, 18083484, 1961470, 176554, 13145, 791, 36, 1

Offset: 0

Igor Victorovich Statsenko, Sep 30 2024

The columns of the triangle T(m,n,k) represent the coefficients of the asymptotic expansion of the higher order exponential integral E(x,m+1,k+2), for m=2, k>=0. For reference see. A163931.

			Triangle starts:
 [0]          1;
 [1]          9,          1;
 [2]         71,         12,          1;
 [3]        580,        119,         15,        1;
 [4]       5104,       1175,        179,       18,        1;
 [5]      48860,      12154,       2070,      251,       21,       1;
 [6]     509004,     133938,      24574,     3325,      335,      24,     1;
 [7]    5753736,    1580508,     305956,    44524,     5000,     431,    27,     1;

Igor Victorovich Statsenko, Relationships of P-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.

Column k: A001706 (k=0), A001712 (k=1), A001717 (k=2), A001722 (k=3), A051525 (k=4), A051546 (k=5), A051561 (k=6).

Cf. A094587 and A173333 for m=0, A376582 for m=1.

T:=(m,n,k)->add(Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!,i=0..n-k):m:=2:seq(seq(T(m,n,k), k=0..n),n=0..10);

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A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

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A001713 Generalized Stirling numbers.

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A001714 Generalized Stirling numbers.

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A196845 Table of elementary symmetric function a_k(3,4,...,n+2) (no 1 and 2).

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A325137 Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows.

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A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)*binomial(n+m+1, n-k-i)*(n + m - k)!/(i + m)!, for m = 2.

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A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)binomial(n+m+1, n-k-i)(n + m - k)!/(i + m)!, for m = 2.