A001713 -id:A001713 - cp's OEIS frontend

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

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.

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

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

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

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

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A001719 Generalized Stirling numbers.

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A001714 Generalized Stirling numbers.

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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.

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