A001706 -id:A001706 - 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

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

A136124 Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 24, 26, 9, 1, 120, 154, 71, 14, 1, 720, 1044, 580, 155, 20, 1, 5040, 8028, 5104, 1665, 295, 27, 1, 40320, 69264, 48860, 18424, 4025, 511, 35, 1, 362880, 663696, 509004, 214676, 54649, 8624, 826, 44, 1, 3628800, 6999840, 5753736, 2655764

Offset: 2

Emeric Deutsch, Dec 23 2007

Sum of entries in row n = n!/2 = A001710(n). T(n,1) = (n-1)! = A000142(n-1). Columns 2,3,4 and 5 yield A001705,A001706,A001707 and A001708, respectively.
See A143491 for the interpretation of these numbers as restricted Stirling numbers of the first kind. See A049444 for a signed version of this array. - Peter Bala, Aug 25 2008
With offset n=0, k=0: triangle T(n,k), read by rows, given by [2,1,3,2,4,3,5,4,6,5,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 29 2011
With offset n=0, k=0: T(n,k) is the number of ways to seat n people at any number of round tables and serve exactly k of the tables water, some number of the remaining tables red wine, and the rest of the tables white wine. - Geoffrey Critzer, Mar 13 2015

			T(6,3)=71 because (-1)^9*[s(6,1)+s(6,2)+s(6,3)]=-(-120+274-225)=71.
Triangle starts:
    1;
    2,   1;
    6,   5,   1;
   24,  26,   9,   1;
  120, 154,  71,  14,   1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Olivier Bodini, Antoine Genitrini, Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.

Cf. A000142, A008275, A001705, A001706, A001707, A001708, A001710.

Cf. A049444, A143491. [Peter Bala, Aug 25 2008]

A136124_row := proc(n) local k,j; `if`(n=0,1,seq((-1)^(n+1-k)*add(stirling1(n+1,j), j=1..k),k=1..n)) end: seq(print(A136124_row(r)),r=1..6); # Peter Luschny, Sep 29 2011
with(combinat): T:=proc(n, k) options operator, arrow: (-1)^(n+k)*(sum(stirling1(n,j),j=1..k)) end proc: for n from 2 to 11 do seq(T(n,k),k=1..n-1) end do; # yields sequence in triangular form

nn = 10; Map[Select[#, # > 0 &] &,Range[0,nn]!CoefficientList[Series[Exp[(2 + y) Log[1/(1 - x)]], {x, 0, nn}], {x,y}]] // Flatten (* Geoffrey Critzer, Mar 13 2015 *)

E.g.f.: Sum[(1/n!)T(n,k)x^n*t^k, k=1..n-1, n>=2]=1/[(1+t)(1-x)^t]-(1+tx)/(1+t). Generating polynomial of row n = t*Product(j+t, j=2..n-1). T(n,k) is the sum of all products of n-k-1 different integers taken from {2,3,...,n-1}. For example, T(6,3) = 2*3 + 2*4 + 2*5 + 3*4 + 3*5 + 4*5 = 71.

A307419 Triangle of harmonic numbers T(n, k) = [t^n] Gamma(n+k+t)/Gamma(k+t) for n >= 0 and 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 11, 9, 1, 0, 50, 71, 18, 1, 0, 274, 580, 245, 30, 1, 0, 1764, 5104, 3135, 625, 45, 1, 0, 13068, 48860, 40369, 11515, 1330, 63, 1, 0, 109584, 509004, 537628, 203889, 33320, 2506, 84, 1, 0, 1026576, 5753736, 7494416, 3602088, 775929, 81900, 4326, 108, 1

Offset: 0

Vladimir Kruchinin, Apr 08 2019

			Triangle starts:
0: [1]
1: [0,       1]
2: [0,       3,       1]
3: [0,      11,       9,       1]
4: [0,      50,      71,      18,       1]
5: [0,     274,     580,     245,      30,      1]
6: [0,    1764,    5104,    3135,     625,     45,     1]
7: [0,   13068,   48860,   40369,   11515,   1330,    63,    1]
8: [0,  109584,  509004,  537628,  203889,  33320,  2506,   84,   1]
9: [0, 1026576, 5753736, 7494416, 3602088, 775929, 81900, 4326, 108, 1]
Col:   A000254, A001706, A001713, A001719, ...

Row sums are A087761.
Columns are A000007, A000254, A001706, A001713, A001719.
Cf. A001008/A002805.

# Note that for n > 16 Maple fails (at least in some versions) to compute the
# terms properly. Inserting 'simplify' or numerical evaluation might help.
A307419Row := proc(n) local ogf, ser; ogf := (n, k) -> GAMMA(n+k+x)/GAMMA(k+x);
ser := (n, k) -> series(ogf(n,k),x,k+2); seq(coeff(ser(n,k),x,k),k=0..n) end: seq(A307419Row(n), n=0..9);
# Alternatively by the egf for column k:
A307419Col := proc(n, len) local f, egf, ser; f := (n,x) -> (log(1-x)/(x-1))^n/n!;
egf := (n,x) -> diff(f(n, x), [x$n]); ser := n -> series(egf(n, x), x, len);
seq(k!*coeff(ser(n), x, k), k=0..len-1) end:
seq(print(A307419Col(k, 10)), k=0..9); # Peter Luschny, Apr 12 2019
T := (n, k) -> add((-1)^(n-j)*binomial(j, k)*Stirling1(n, j)*k^(j-k), j = k..n):
seq(seq(T(n,k), k = 0..n), n = 0..9); # Peter Luschny, Jun 09 2022

f[n_, x_] := f[n, x] = D[(Log[1 - x]/(x - 1))^n/n!, {x, n}];
T[n_, k_] := (n - k)! SeriesCoefficient[f[k, x], {x, 0, n - k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 13 2019 *)

T(n,k):=n!*sum((binomial(k+i-1,i)*abs(stirling1(n-i,k)))/(n-i)!,i,0,n-k);

Maxima
```
taylor((1-t)^(-x/(1-t)),t,0,7,x,0,7);
```

T(n,k):=coeff(taylor(gamma(n+k+t)/gamma(k+t),t,0,10),t,k);

T(n, k) = n!*sum(i=0, n-k, abs(stirling(n-i, k, 1))*binomial(i+k-1, i)/(n-i)!); \\ Michel Marcus, Apr 13 2019

E.g.f.: A(t,x) = (1-t)^(-x/(1-t)).
T(n, k) = n!*Sum_{L1+L2+...+Lk=n} H(L1)H(L2)...H(Lk) with Li > 0, where H(n) are the harmonic numbers A001008.
T(n, k) = n!*Sum_{i=0..n-k} abs(Stirling1(n-i, k))/(n-i)!*binomial(i+k-1, i).
T(n, k) = k! [x^k] (d^n/dx^n) ((log(1-x)/(x-1))^n/n!), the e.g.f. for column k where Col(k) = [T(n+k, k) for n = 0, 1, 2, ...]. - Peter Luschny, Apr 12 2019
T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(j, k)*Stirling1(n, j)*k^(j-k). - Peter Luschny, Jun 09 2022

A347989 a(n) = [x^n] (2n)! Sum_{k=0..2*n} binomial(x+k,k).

Original entry on oeis.org

1, 5, 71, 1665, 54649, 2310945, 119753843, 7353403057, 522289211873, 42137920501677, 3807384320667135, 380929847762489025, 41811136672902061321, 4995760464106519955705, 645541681316043216096315, 89705032647088734873129825, 13340173206548155385625683265, 2114001534402053456524492822485

Offset: 0

Seiichi Manyama, Sep 23 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..325

Cf. A001706, A001707, A001708, A001709, A008275, A143491, A347987.

a(n) = (2*n)!*polcoef(sum(k=n, 2*n, binomial(x+k, k)), n);

a(n) = (2*n)!*sum(k=n, 2*n, (2*n+1-k)*abs(stirling(k, n, 1))/k!);

a(n) = (2*n)! * Sum_{k=n..2*n} (2*n+1-k) * |Stirling1(k,n)|/k!.
a(n) = [x^(2*n)] ((2*n)!/n!) * (-log(1 - x))^n/(1 - x)^2.
From Vaclav Kotesovec, Sep 23 2021, updated May 14 2025: (Start)
a(n) = [x^n] Gamma(2*n + x + 2) / Gamma(x + 2).
a(n) ~ c * d^n * (n-1)!, where d = 8*w^2/(2*w-1), w = -LambertW(-1,-exp(-1/2)/2) and c = w^2 * sqrt(2) / (sqrt(w-1)*Pi) = 1.5967712192197964362930380385801737624829174112909160160618... (End)

A193446 a(n) = n! * Sum_{k=1..n-1} H(k)*H(n-k) for n>=2, where H(n) is the n-th harmonic number.

Original entry on oeis.org

2, 18, 142, 1160, 10208, 97720, 1018008, 11507472, 140581872, 1848236544, 26041957632, 391738883328, 6268657962240, 106361504663040, 1907768564282880, 36075270482058240, 717379367864693760, 14967427450111488000, 326956068509511168000, 7463341244426167296000

Offset: 2

Paul D. Hanna, Jul 26 2011

nonn

A self-convolution of the harmonic numbers.

			E.g.f.: log(1-x)^2/(1-x)^2 = 2*x^2/2! + 18*x^3/3! + 142*x^4/4! + 1160*x^5/5! + ...
Illustration of initial terms:
a(2) = 2!*(1*1) = 2;
a(3) = 3!*(1*(1+1/2) + (1+1/2)*1) = 18;
a(4) = 4!*(1*(1+1/2+1/3) + (1+1/2)*(1+1/2) + (1+1/2+1/3)*1) = 142;
a(5) = 5!*(1*(1+1/2+1/3+1/4) + (1+1/2)*(1+1/2+1/3) + (1+1/2+1/3)*(1+1/2) + (1+1/2+1/3+1/4)*1) = 1160; ...

Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A001008, A002805, A001706, A193445.

a[n_] := n! * Sum[HarmonicNumber[k] * HarmonicNumber[n-k], {k, 1, n-1}]; Array[a, 20, 2] (* Amiram Eldar, Aug 18 2023 *)

{a(n)=if(n<2,0,n!*sum(k=1,n,sum(i=1,k,1/i)*sum(j=1,n-k,1/j)))}

{a(n)=if(n<0,0,n!*polcoeff(log(1-x+x*O(x^n))^2/(1-x+x*O(x^n))^2,n))}

E.g.f.: log(1 - x)^2 / (1 - x)^2.

a(n) = n! * Sum_{k=1..n-1} (Sum_{i=1..k} 1/i)*(Sum_{j=1..n-k} 1/j) for n>=2.

a(21) from Amiram Eldar, Aug 18 2023

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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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.