cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A060096 Numerator of coefficients of Euler polynomials (rising powers).

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, 0, -3, 1, 0, 1, 0, -2, 1, -1, 0, 5, 0, -5, 1, 0, -3, 0, 5, 0, -3, 1, 17, 0, -21, 0, 35, 0, -7, 1, 0, 17, 0, -28, 0, 14, 0, -4, 1, -31, 0, 153, 0, -63, 0, 21, 0, -9, 1, 0, -155, 0, 255, 0, -126, 0, 30, 0, -5, 1, 691, 0, -1705, 0, 2805, 0, -231, 0, 165, 0, -11, 1, 0, 2073, 0, -3410, 0, 1683, 0, -396
Offset: 0

Views

Author

Wolfdieter Lang, Mar 29 2001

Keywords

Comments

From S. Roman, The Umbral Calculus (see the reference in A048854), p. 101, (4.2.10) (corrected): E(n,x)= sum(sum(binomial(n,m)*((-1/2)^j)*j!*S2(n-m,j),j=0..k)*x^m,m=0..n), with S2(n,m)=A008277(n,m) and S2(n,0)=1 if n=0 else 0 (Stirling2).
From Wolfdieter Lang, Oct 31 2011: (Start)
This is the Sheffer triangle (2/(exp(x)+1),x) (which would be called in the above mentioned S. Roman reference Appell for (exp(t)+1)/2) (see p. 27).
The e.g.f. for the row sums is 2/(1+exp(-x)). The row sums look like A198631(n)/A006519(n+1), n>=0.
The e.g.f. for the alternating row sums is 2/(exp(x)*(exp(x)+1)). These sums look like (-1)^n*A143074(n)/ A006519(n+1).
The e.g.f. for the a-sequence of this Sheffer array is 1. The z-sequence has e.g.f. (1-exp(x))/(2*x). This z-sequence is -1/(2*A000027(n))=-1/(2*(n+1)) (see the link under A006232 for the definition of a- and z-sequences). This leads to the recurrences given below.
The alternating power sums for the first n positive integers are given by sum((-1)^(n-j)*j^k,j=1..n) = (E(k, x=n+1)+(-1)^n*E(k, x=0))/2, k>=1, n>=1,with the row polynomials E(n, x)(see the Abramowitz-Stegun reference, p. 804, 23.1.4, and an addendum in the W. Lang link under A196837).
(End)

Examples

			n\m  0    1    2    3    4    5    6    7  8  ...
0:   1
1:  -1    1
2:   0   -1    1
3:   1    0   -3    1
4:   0    1    0   -2    1
5:  -1    0    5    0   -5    1
6:   0   -3    0    5    0   -3    1
7:  17    0  -21    0   35    0   -7    1
8:   0   17    0  -28    0   14    0   -4  1
...
The rational triangle a(n,m)/A060097(n,m) starts
n\m  0    1    2    3    4    5    6    7  8  ...
0:   1
1: -1/2   1
2:   0   -1    1
3:  1/4   0  -3/2   1
4:   0    1    0   -2    1
5: -1/2   0   5/2   0  -5/2   1
6:   0   -3    0    5    0   -3    1
7: 17/8   0 -21/2   0  35/4   0  -7/2   1
8:   0   17    0  -28    0   14    0   -4  1
...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 20, equations 20:4:1 - 20:4:8 at pages 177-178.

Links

  • 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].

Crossrefs

Cf. A060097.

Programs

  • Maple
    A060096 := proc(n,m) coeff(euler(n,x),x,m) ; numer(%) ;end proc:
seq(seq(A060096(n,m),m=0..n),n=0..12) ; # R. J. Mathar, Dec 21 2010
  • Mathematica
    Numerator[Flatten[Table[CoefficientList[EulerE[n, x], x], {n, 0, 12}]]] (* Jean-François Alcover, Apr 29 2011 *)

Formula

E(n, x)= sum((a(n, m)/b(n, m))*x^m, m=0..n), denominators b(n, m)= A060097(n, m).
From Wolfdieter Lang, Oct 31 2011: (Start)
E.g.f. for E(n, x) is 2*exp(x*z)/(exp(z)+1).
E.g.f. of column no. m, m>=0, is 2*x^{m+1}/(m!*(exp(x)+1)).
Recurrences for E(n,m):=a(n,m)/A060097(n,m) from the Sheffer a-and z-sequence:
E(n,m)=(n/m)*E(n-1,m-1), n>=1,m>=1.
E(n,0)=-n*sum(E(n-1,j)/(2*(j+1)),j=0..n-1), n>=1, E(0,0)=1.
(see the Sheffer comments above).
(End)
E(n,m) = binomial(n,m)*sum(((-1)^j)*j!*S2(n-m,j)/2^j ,j=0..n-m), 0<=m<=n, with S2 given by A008277. From S. Roman, The umbral calculus, reference under A048854, eq. (4.2.10), p. 101, with a=1, and a misprint corrected: replace 1/k! by binomial(n,k) (also in the two preceding formulas). - Wolfdieter Lang, Nov 03 2011
The first (m=0) column of the rational triangle is conjectured to be E(n,0) = ((-1)^n)*A198631(n) / A006519(n+1). See also the first column shown in A209308 (different signs). - Wolfdieter Lang, Jun 15 2015

Extensions

Table rewritten by Wolfdieter Lang, Oct 31 2011

A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).

Original entry on oeis.org

1, 6, -1, 42, -14, 1, 336, -168, 24, -1, 3024, -2016, 432, -36, 1, 30240, -25200, 7200, -900, 50, -1, 332640, -332640, 118800, -19800, 1650, -66, 1, 3991680, -4656960, 1995840, -415800, 46200, -2772, 84, -1, 51891840, -69189120
Offset: 0

Views

Author

Wolfdieter Lang, Jun 19 2001

Keywords

Comments

The row polynomials s(n,x) := n!*L(n,5,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^6. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials sum(|A008297(n,m)|*(-x)^m, m=1..n), n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).
These polynomials appear in the radial part of the l=2 (d-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148-A062152. Row sums (signed) give A062191; row sums (unsigned) give A062192.
The unsigned version of this triangle is the triangle of unsigned 3-Lah numbers A143498. - Peter Bala, Aug 25 2008

Examples

			Triangle begins:
  {1};
  {6, -1};
  {42, -14, 1};
  {336, -168, 24, -1};
  ...
2!*L(2, 5, x) = 42-14*x+x^2.

References

  • A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.

Links

Crossrefs

Cf. A021009, A062137, A062139, A062140, A066667.
For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148, A062149, A062150, A062151, A062152.
Row sums (signed) give A062191, row sums (unsigned) give A062192.
Cf. A143498.

Programs

  • Mathematica
    Flatten[Table[((-1)^m)*n!*Binomial[n+5,n-m]/m!,{n,0,8},{m,0,n}]] (* Indranil Ghosh, Feb 24 2017 *)
  • PARI
    tabl(nn) = {for (n=0, nn, for (m=0, n, print1(((-1)^m)*n!*binomial(n+5, n-m)/m!, ", "); ); print(); ); } \\ Indranil Ghosh, Feb 24 2017
  • PARI
    row(n) = Vecrev(n!*pollaguerre(n, 5)); \\ Michel Marcus, Feb 06 2021
  • Python
    import math
f=math.factorial
def C(n, r):return f(n)//f(r)//f(n-r)
i=-1
for n in range(26):
    for m in range(n+1):
        i += 1
        print(str(i)+" "+str(((-1)**m)*f(n)*C(n+5, n-m)//f(m))) # Indranil Ghosh, Feb 24 2017

Formula

T(n, m) = ((-1)^m)*n!*binomial(n+5, n-m)/m!.
E.g.f. for m-th column: ((-x/(1-x))^m)/(m!*(1-x)^6), m >= 0.

A049458 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -3, 1, 12, -7, 1, -60, 47, -12, 1, 360, -342, 119, -18, 1, -2520, 2754, -1175, 245, -25, 1, 20160, -24552, 12154, -3135, 445, -33, 1, -181440, 241128, -133938, 40369, -7140, 742, -42, 1, 1814400, -2592720, 1580508, -537628
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,m)= ^3P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(3+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1.
In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer polynomials for (exp(3*t),exp(t)-1).
See A143492 for the unsigned version of this array and A143495 for the inverse. - Peter Bala, Aug 25 2008

Examples

			1;
-3, 1;
12, -7, 1;
-60, 47, -12, 1;
360, -342, 119, -18, 1;
s(2,x) = 12-7*x+x^2. S1(2,x) = -x+x^2 (Stirling1 polynomial).

References

  • Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Links

Crossrefs

Unsigned column sequences are: A001710-A001714. Row sums (signed triangle): (n+1)!*(-1)^n. Row sums (unsigned triangle): A001715(n+3).
Cf. A000035 A084938.
Cf. A094645, A094646.
A143492, A143495. - Peter Bala, Aug 25 2008

Programs

  • Haskell
    a049458 n k = a049458_tabl !! n !! k
a049458_row n = a049458_tabl !! n
a049458_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 3)
-- Reinhard Zumkeller, Mar 11 2014
  • Maple
    A049458_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+3, n)), x, k), k=0..n): seq(print(A049458_row(n)),n=0..8); # Peter Luschny, May 16 2013
  • Mathematica
    t[n_, k_] := (-1)^(n - k)*Coefficient[ Pochhammer[x + 3, n], x, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 17 2013, after Peter Luschny *)

Formula

a(n, m)= a(n-1, m-1) - (n+2)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
Triangle (signed) = [ -3, -1, -4, -2, -5, -3, -6, -4, -7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [3, 1, 4, 2, 5, 3, 6, 4, 7, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143492).
E.g.f.: (1+y)^(x-3). - Vladeta Jovovic, May 17 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 T(n,i) = f(n,i,3), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Extensions

Second formula corrected by Philippe Deléham, Nov 09 2008

A051338 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -6, 1, 42, -13, 1, -336, 146, -21, 1, 3024, -1650, 335, -30, 1, -30240, 19524, -5000, 635, -40, 1, 332640, -245004, 74524, -11985, 1075, -51, 1, -3991680, 3272688, -1139292, 218344, -24885, 1687, -63, 1, 51891840, -46536624, 18083484, -3977764, 541849, -46816, 2506, -76, 1
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,m)= ^6P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(6+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(6*t),exp(t)-1).

Examples

			{1}; {-6,1}; {42,-13,1}; {-336,146,-21,1}; ... s(2,x)= 42-13*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

References

  • Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Links

Crossrefs

Unsigned m=0 column sequence is: A001725. Row sums (signed triangle): A001720(n+4)*(-1)^n. Row sums (unsigned triangle): A001730(n+6).
Cf. A000035 A084938.

Programs

  • Haskell
    a051338 n k = a051338_tabl !! n !! k
a051338_row n = a051338_tabl !! n
a051338_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 6)
-- Reinhard Zumkeller, Mar 11 2014
  • Mathematica
    t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*Pochhammer[6, k]*StirlingS1[n - k, i], {k, 0, n - i}]; Flatten[Table[t[n, i], {n, 0, 8}, {i, 0, n}]][[1 ;; 45]] (* Jean-François Alcover, Jun 01 2011, after Milan Janjic *)

Formula

a(n, m)= a(n-1, m-1) - (n+5)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^6).
Triangle (signed) = [ -6, -1, -7, -2, -8, -3, -9, -4, -10, ...] DELTA A000035; triangle (unsigned) = [6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.
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 T(n,i) = f(n,i,6), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

A051339 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -7, 1, 56, -15, 1, -504, 191, -24, 1, 5040, -2414, 431, -34, 1, -55440, 31594, -7155, 805, -45, 1, 665280, -434568, 117454, -16815, 1345, -57, 1, -8648640, 6314664, -1961470, 336049, -34300, 2086, -70, 1, 121080960, -97053936, 33775244, -6666156, 816249, -63504, 3066, -84, 1
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,m)= ^7P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(7+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(7*t),exp(t)-1).

Examples

			{1}; {-7,1}; {56,-15,1}; {-504,191,-24,1}; ... s(2,x)= 56-15*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

Links

Crossrefs

The first (m=0) column sequence is A001730. Row sums (signed triangle): A001725(n+5)*(-1)^n. Row sums (unsigned triangle): A049388(n).
Cf. A000035 A084938.

Programs

  • Haskell
    a051339 n k = a051339_tabl !! n !! k
a051339_row n = a051339_tabl !! n
a051339_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 7)
-- Reinhard Zumkeller, Mar 11 2014
  • Mathematica
    a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^7, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

Formula

a(n, m)= a(n-1, m-1) - (n+6)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^7).
Triangle (signed) = [ -7, -1, -8, -2, -9, -3, -10, -4, -11, -5, ...] DELTA A000035; triangle (unsigned) = [7, 1, 8, 2, 9, 3, 10, 4, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.
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 T(n,i) = f(n,i,7), for n=1,2,...;i=0...n. [From Milan Janjic, Dec 21 2008]

A051379 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -8, 1, 72, -17, 1, -720, 242, -27, 1, 7920, -3382, 539, -38, 1, -95040, 48504, -9850, 995, -50, 1, 1235520, -725592, 176554, -22785, 1645, -63, 1, -17297280, 11393808, -3197348, 495544, -45815, 2527, -77, 1, 259459200, -188204400, 59354028, -10630508, 1182769, -83720, 3682, -92, 1
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,m)= ^8P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(8+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(8*t),exp(t)-1).

Examples

			{1}; {-8,1}; {72,-17,1}; {-720,242,-27,1}; ... s(2,x)=72-17*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

Links

Crossrefs

The first (m=0) column sequence is: A049388. Row sums (signed triangle): A001730(n+6)*(-1)^n. Row sums (unsigned triangle): A049389(n).
Cf. A000035 A084938.

Programs

  • Haskell
    a051379 n k = a051379_tabl !! n !! k
a051379_row n = a051379_tabl !! n
a051379_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 8)
-- Reinhard Zumkeller, Mar 12 2014
  • Mathematica
    a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^8, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

Formula

a(n, m)= a(n-1, m-1) - (n+7)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^8).
Triangle (signed) = [ -8, -1, -9, -2, -10, -3, -11, -4, -12, ...] DELTA A000035; triangle (unsigned) = [8, 1, 9, 2, 10, 3, 11, 4, 12, 5, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.
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 T(n,i) = f(n,i,8), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Extensions

Typo fixed in data by Reinhard Zumkeller, Mar 12 2014

A132062 Sheffer triangle (1,1-sqrt(1-2*x)). Extended Bessel triangle A001497.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 15, 15, 6, 1, 0, 105, 105, 45, 10, 1, 0, 945, 945, 420, 105, 15, 1, 0, 10395, 10395, 4725, 1260, 210, 21, 1, 0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1, 0, 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1, 0
Offset: 0

Views

Author

Wolfdieter Lang Sep 14 2007

Keywords

Comments

This is a Jabotinsky type exponential convolution triangle related to A001147 (double factorials). For Jabotinsky type triangles See the D. E. Knuth reference given under A039692.
The subtriangle (n>=m>=1) is A001497(n,m) (Bessel).
For the combinatorial interpretation in terms of unordered forests of increasing plane trees see the W. Lang comment and example under A001497.
This is a special type of Sheffer triangle. See the S. Roman reference given under A048854 (the notation here differs).
This triangle (or the A001497 subtriangle) appears as generalized Stirling numbers of the second kind, S2p(-1,n,m):=S2(-k;m,m)*(-1)^(n-m) for k=1, eqs. (27)-(29) of the W. Lang reference.
Also the Bell transform of the double factorial of odd numbers A001147. For the Bell transform of the double factorial of even numbers A000165 see A039683. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015

Examples

			[1]
[0,      1]
[0,      1,      1]
[0,      3,      3,     1]
[0,     15,     15,     6,     1]
[0,    105,    105,    45,    10,    1]
[0,    945,    945,   420,   105,   15,   1]
[0,  10395,  10395,  4725,  1260,  210,  21,  1]
[0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1]

References

  • Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From N. J. A. Sloane, Sep 15 2012
  • Steven Roman, The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, 1984. (p. 78) [Emanuele Munarini, Oct 10 2017]

Links

Crossrefs

Columns m=1: A001147.
Row sums give [1, A001515]. Alternating row sums give [1, -A000806].
Cf. A122850. - R. J. Mathar, Mar 20 2009
Cf. A039683, A264428.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
BellMatrix(n -> doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016
# Alternative:
egf := exp(y*(1 - sqrt(1 - 2*x))): serx := series(egf, x, 12):
coefx := n -> n!*coeff(serx, x, n): row := n -> seq(coeff(coefx(n), y, k), k = 0..n): for n from 0 to 8 do row(n) od;  # Peter Luschny, Apr 25 2024
  • Mathematica
    Table[If[k <= n, Binomial[2n-2k,n-k] Binomial[2n-k-1,k-1] (n-k)!/2^(n-k), 0], {n, 0, 6}, {k, 0, n}] // Flatten (* Emanuele Munarini, Oct 10 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[(2#-1)!!&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
  • Sage
    # uses[bell_transform from A264428]
def A132062_row(n):
    a = sloane.A001147
    dblfact = a.list(n)
    return bell_transform(n, dblfact)
[A132062_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015

Formula

a(n,m)=0 if n
E.g.f. m-th column ((x*f2p(1;x))^m)/m!, m>=0. with f2p(1;x):=1-sqrt(1-2*x)= x*c(x/2) with the o.g.f.of A000108 (Catalan).
From Emanuele Munarini, Oct 10 2017: (Start)
a(n,k) = binomial(2*n-2*k,n-k)*binomial(2*n-k-1,k-1)*(n-k)!/2^(n-k).
The row polynomials p_n(x) (studied by Carlitz) satisfy the recurrence: p_{n+2}(x) - (2*n+1)*p_{n+1}(x) - x^2*p_n(x) = 0. (End)
T(n, k) = n! [y^k] [x^n] exp(y*(1 - sqrt(1 - 2*x))). - Peter Luschny, Apr 25 2024

A049459 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -4, 1, 20, -9, 1, -120, 74, -15, 1, 840, -638, 179, -22, 1, -6720, 5944, -2070, 355, -30, 1, 60480, -60216, 24574, -5265, 625, -39, 1, -604800, 662640, -305956, 77224, -11515, 1015, -49, 1, 6652800, -7893840, 4028156, -1155420, 203889
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,m)= ^4P_n^m in the notation of the given reference with a(0,0) := 1.
The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(4+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1.
In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(4*t),exp(t)-1).
See A143493 for the unsigned version of this array and A143496 for the inverse. - Peter Bala, Aug 25 2008

Examples

			   1;
  -4,    1;
  20,   -9,   1;
-120,   74, -15,   1;
840, -638, 179, -22, 1;

References

  • Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Links

Crossrefs

Unsigned column sequences are: A001715-A001719. Cf. A008275 (Stirling1 triangle), A049458, A049460. Row sums (signed triangle): A001710(n+2)*(-1)^n. Row sums (unsigned triangle): A001720(n+4).
Cf. A000035 A084938.
A143493, A143496. - Peter Bala, Aug 25 2008

Programs

  • Haskell
    a049459 n k = a049459_tabl !! n !! k
a049459_row n = a049459_tabl !! n
a049459_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 4)
-- Reinhard Zumkeller, Mar 11 2014
  • Maple
    A049459_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+4, n)), x, k), k=0..n): seq(print(A049459_row(n)),n=0..8); # Peter Luschny, May 16 2013
  • Mathematica
    a[n_, m_] /; 0 <= m <= n := a[n, m] = a[n-1, m-1] - (n+3)*a[n-1, m];
a[n_, m_] /; n < m = 0;
a[_, -1] = 0; a[0, 0] = 1;
Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

Formula

a(n, m)= a(n-1, m-1) - (n+3)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
Triangle (signed) = [ -4, -1, -5, -2, -6, -3, -7, -4, -8, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [4, 1, 5, 2, 6, 3, 7, 4, 8, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143493).
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 T(n,i) = f(n,i,4), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Extensions

Second formula corrected by Philippe Deléham, Nov 09 2008

A049460 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -5, 1, 30, -11, 1, -210, 107, -18, 1, 1680, -1066, 251, -26, 1, -15120, 11274, -3325, 485, -35, 1, 151200, -127860, 44524, -8175, 835, -45, 1, -1663200, 1557660, -617624, 134449, -17360, 1330, -56, 1, 19958400, -20355120, 8969148, -2231012, 342769, -33320, 2002, -68, 1
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,m)= ^5P_n^m in the notation of the given reference with a(0,0) := 1.
The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(5+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1.
In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(5*t),exp(t)-1).

Examples

			{1}; {-5,1}; {30,-11,1}; {-210,107,-18,1}; ... s(2,x)= 30-11*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

Links

Crossrefs

Unsigned column sequences are: A001720-A001724. Row sums (signed triangle): A001715(n+3)*(-1)^n. Row sums (unsigned triangle): A001725(n+5).
Cf. A000035 A084938.

Programs

  • Haskell
    a049460 n k = a049460_tabl !! n !! k
a049460_row n = a049460_tabl !! n
a049460_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 5)
-- Reinhard Zumkeller, Mar 11 2014
  • Mathematica
    a[n_, m_] := Pochhammer[m+1, n-m] SeriesCoefficient[Log[1+x]^m/(1+x)^5, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

Formula

a(n, m)= a(n-1, m-1) - (n+4)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
Triangle (signed) = [ -5, -1, -6, -2, -7, -3, -8, -4, -9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...]; triangle (unsigned) = [5, 1, 6, 2, 7, 3, 8, 4, 9, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938.
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 T(n,i) = f(n,i,5), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Extensions

Second formula corrected by Philippe Deléham, Nov 10 2008

A051380 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -9, 1, 90, -19, 1, -990, 299, -30, 1, 11880, -4578, 659, -42, 1, -154440, 71394, -13145, 1205, -55, 1, 2162160, -1153956, 255424, -30015, 1975, -69, 1, -32432400, 19471500, -4985316, 705649, -59640, 3010, -84, 1, 518918400, -343976400, 99236556, -16275700, 1659889, -107800, 4354, -100, 1
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,m)= ^9P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(9+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(9*t),exp(t)-1).

Examples

			{1}; {-9,1}; {90,-19,1}; {-990,299,-30,1}; ... s(2,x)= 90-19*x+x^2; S1(2,x)= -x+x^2 (Stirling1).

Links

Crossrefs

The first (m=0) column sequence is: A049389. Row sums (signed triangle): A049388(n)*(-1)^n. Row sums (unsigned triangle): A049398(n).
Cf. A000035 A084938.

Programs

  • Haskell
    a051380 n k = a051380_tabl !! n !! k
a051380_row n = a051380_tabl !! n
a051380_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 9)
-- Reinhard Zumkeller, Mar 12 2014
  • Mathematica
    a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^9, {x, 0, n}];
Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2019 *)

Formula

a(n, m)= a(n-1, m-1) - (n+8)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n
E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^9).
Triangle (signed) = [ -9, -1, -10, -2, -11, -3, -12, -4, -13, ...] DELTA A000035; triangle (unsigned) = [9, 1, 10, 2, 11, 3, 12, 4, 13, 5, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.
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 T(n,i) = f(n,i,9), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008
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