A008969 -id:A008969 - cp's OEIS frontend

A001237 Differences of reciprocals of unity.

31, 3661, 1217776, 929081776, 1413470290176, 3878864920694016, 17810567950611972096, 129089983180418186674176, 1409795030885143760732160000, 22335321387514981111936450560000, 497400843208278958640564703068160000, 15161356456130244705175927906904309760000

Offset: 1

N. J. A. Sloane

nonn

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
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).

Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Column 4 in triangle A008969.

a[n_] := -(Factorial[n + 1]^4)*Sum[(-1)^k Binomial[n + 1, k]/k^4, {k, 1, n + 1}];Table[a[n],{n,14}] (* James C. McMahon, Dec 12 2023 *)

a(n)=-(n+1)!^4*sum(k=1,n+1,(-1)^k*binomial(n+1,k)/k^4) \\ Charles R Greathouse IV, Mar 29 2012

a(n) = (n + 1)!^4/480*(20*Psi(n + 2)^4 + 80*gamma*Psi(n + 2)^3 - 120*Psi(n + 2)^2*Psi(1, n + 2) + 20*Pi^2*Psi(n + 2)^2 + 120*gamma^2*Psi(n + 2)^2 - 240*gamma*Psi(n + 2)*Psi(1, n + 2) + 80*Psi(n + 2)*Psi(2, n + 2) + 60*Psi(1, n + 2)^2 + 40*gamma*Pi^2*Psi(n + 2) + 160*Zeta(3)*Psi(n + 2) + 80*gamma^3*Psi(n + 2) - 20*Pi^2*Psi(1, n + 2) - 120*gamma^2*Psi(1, n + 2) + 80*gamma*Psi(2, n + 2) - 20*Psi(3, n + 2) + 160*gamma*Zeta(3) + 3*Pi^4 + 20*gamma^4 + 20*gamma^2*Pi^2). - Vladeta Jovovic, Aug 10 2002
a(n) = (n+1)!^4 * Sum_{i=1..n+1} Sum_{j=1..i} Sum_{k=1..j} Sum_{l=1..k} 1/(ijkl).
a(n) = (n+1)!^4 * Sum_{k=1..n+1} (-1)^(k+1)*C(n+1,k)/k^4. - Sean A. Irvine, Mar 29 2012

More terms from Vladeta Jovovic, Aug 10 2002

a(11)-a(12) from James C. McMahon, Dec 12 2023

A111887 Seventh column of triangle A112492 (inverse scaled Pochhammer symbols).

Original entry on oeis.org

1, 13068, 104587344, 673781602752, 3878864920694016, 21006340945438768128, 110019668725577574273024, 565858042127972959667208192, 2882220940619488483325345857536, 14605752814655604919042956624396288

Offset: 0

Wolfdieter Lang, Sep 12 2005

Also continuation of family of differences of reciprocals of unity. See A001242, A111886 and triangle A008969.

G. C. Greubel, Table of n, a(n) for n = 0..250
Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Also right-hand column 6 in triangle A008969.

Cf. A001242, A111886, A112492.

A111887:= func< n | (-1)*Factorial(7)^n*(&+[(-1)^j*Binomial(7,j)/j^n : j in [1..7]]) >;
[A111887(n): n in [0..30]]; // G. C. Greubel, Jul 24 2023

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, (k+1)^(n-k)*T[n-1,k-1] + k!*T[n-1,k]]; (* T = A112492 *)
Table[T[n+6,6], {n,0,30}] (* G. C. Greubel, Jul 24 2023 *)

a(n) = -((7!)^n)*sum(j=1, 7, ((-1)^j)*binomial(7, j)/j^n); \\ Michel Marcus, Apr 28 2020

@CachedFunction
def T(n,k): # T = A112492
    if (k==0 or k==n): return 1
    else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
def A111887(n): return T(n+6,6)
[A111887(n) for n in range(31)] # G. C. Greubel, Jul 24 2023

G.f.: 1/Product_{j=1..7} 1-7!*x/j.
a(n) = -((7!)^n) * Sum_{j=1..7} (-1)^j*binomial(7, j)/j^n, n>=0.
a(n) = A112492(n+6, 7), n>=0.

A111888 Eighth column of triangle A112492 (inverse scaled Pochhammer symbols).

Original entry on oeis.org

1, 109584, 7245893376, 381495483224064, 17810567950611972096, 778101042571221893382144, 32762625292956765972873609216, 1351813956241264848815287984717824

Offset: 0

Wolfdieter Lang, Sep 12 2005

Also continuation of family of Differences of reciprocals of unity. See A001242, A111887 and triangle A008969.

Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Also right-hand column 7 in triangle A008969.

Cf. A001242, A111887, A112492.

A111888:= func< n | (-1)*Factorial(8)^n*(&+[(-1)^j*Binomial(8,j)/j^n : j in [1..8]]) >;
[A111888(n): n in [0..30]]; // G. C. Greubel, Jul 24 2023

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, (k+1)^(n-k)*T[n-1,k-1] +k!*T[n-1,k]]; (* T = A112492 *)
Table[T[n+7,7], {n,0,30}] (* G. C. Greubel, Jul 24 2023 *)

a(n) = -((8!)^n)*sum(j=1, 8, ((-1)^j)*binomial(8, j)/j^n); \\ Michel Marcus, Apr 28 2020

@CachedFunction
def T(n,k): # T = A112492
    if (k==0 or k==n): return 1
    else: return (k+1)^(n-k)*T(n-1,k-1) + factorial(k)*T(n-1,k)
def A111888(n): return T(n+7,7)
[A111888(n) for n in range(31)] # G. C. Greubel, Jul 24 2023

G.f.: 1/Product_{j=1..8} 1-8!*x/j.
a(n) = -((8!)^n) * Sum_{j=1..8} (-1)^j*binomial(8, j)/j^n, n>=0.
a(n) = A112492(n+7, 8), n>=0.

A103879 Square array T(n,k) read by antidiagonals: numerators of Stirling numbers of first kind with negative argument S1(-n,k), n,k>=0.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -3, 1, 0, 1, -11, 7, -1, 0, 1, -25, 85, -15, 1, 0, 1, -137, 415, -575, 31, -1, 0, 1, -49, 12019, -5845, 3661, -63, 1, 0, 1, -121, 13489, -874853, 76111, -22631, 127, -1, 0, 1, -761, 726301, -336581, 58067611, -952525, 137845, -255, 1

Offset: 0

Ralf Stephan, Feb 20 2005

			1, 0, 0, 0, 0, 0,
1, -1, 1, -1, 1, -1,
1/2, -3/4, 7/8, -15/16, 31/32, -63/64,
1/6, -11/36, 85/216, -575/1296, 3661/7776, -22631/46656,
1/24,-25/288,415/3456,-5845/41472,76111/497664,-952525/5971968,

D. Loeb, Generalization of the Stirling numbers of first kind

Denominators are in A103880. Cf. A008969.

T(n,k)=numerator(1/n!*polcoeff(Ser(1/prod(i=1,n,1+x/i)),k))

T(n, k) = (-1)^(k+1) * Sum[i=1..n, C(n, i)*(-1)^i*i^(-k) ].

G.f. of n-th row: 1/n! * 1/Prod[i=1..n, 1+x/i ].

A103880 Square array T(n,k) read by antidiagonals: denominators of Stirling numbers of first kind with negative argument S1(-n,k), n,k>=0.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 24, 36, 8, 1, 1, 120, 288, 216, 16, 1, 1, 720, 7200, 3456, 1296, 32, 1, 1, 5040, 14400, 432000, 41472, 7776, 64, 1, 1, 40320, 235200, 2592000, 25920000, 497664, 46656, 128, 1, 1, 362880, 11289600, 889056000, 51840000

Offset: 0

Ralf Stephan, Feb 20 2005

			1, 0, 0, 0, 0, 0,
1, -1, 1, -1, 1, -1,
1/2, -3/4, 7/8, -15/16, 31/32, -63/64,
1/6, -11/36, 85/216, -575/1296, 3661/7776, -22631/46656,
1/24,-25/288,415/3456,-5845/41472,76111/497664,-952525/5971968,

D. Loeb, Generalization of the Stirling numbers of first kind

Numerators are in A103879. Cf. A008969.

T(n,k)=denominator(1/n!*polcoeff(Ser(1/prod(i=1,n,1+x/i)),k))

T(n, k) = (-1)^(k+1) * Sum[i=1..n, C(n, i)*(-1)^i*i^(-k) ].

G.f. of n-th row: 1/n! * 1/Prod[i=1..n, 1+x/i ].

cp's OEIS Frontend

A001237 Differences of reciprocals of unity.

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A111887 Seventh column of triangle A112492 (inverse scaled Pochhammer symbols).

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A111888 Eighth column of triangle A112492 (inverse scaled Pochhammer symbols).

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A103879 Square array T(n,k) read by antidiagonals: numerators of Stirling numbers of first kind with negative argument S1(-n,k), n,k>=0.

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A103880 Square array T(n,k) read by antidiagonals: denominators of Stirling numbers of first kind with negative argument S1(-n,k), n,k>=0.

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