A347987 -id:A347987 - cp's OEIS frontend

A054651 Triangle T(n,k) read by rows giving coefficients in expansion of n! * Sum_{i=0..n} C(x,i) in descending powers of x.

1, 1, 1, 1, 1, 2, 1, 0, 5, 6, 1, -2, 11, 14, 24, 1, -5, 25, 5, 94, 120, 1, -9, 55, -75, 304, 444, 720, 1, -14, 112, -350, 1099, 364, 3828, 5040, 1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320, 1, -27, 366, -2646, 12873, -31563, 79064, 34236, 270576, 362880

Offset: 0

N. J. A. Sloane, Apr 17 2000

Apparently A190782 with reversed rows. - Mathew Englander, May 17 2014

			The first few polynomials are:
  1, 1+x, 2+x+x^2, 6+5*x+x^3, 24+14*x+11*x^2-2*x^3+x^4, ...
So the triangle begins:
  1;
  1,   1;
  1,   1,   2;
  1,   0,   5,     6;
  1,  -2,  11,    14,   24;
  1,  -5,  25,     5,   94,   120;
  1,  -9,  55,   -75,  304,   444,   720;
  1, -14, 112,  -350, 1099,   364,  3828,  5040;
  1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened

T(2*n,n) gives A347987.

Cf. A054649, A054655, A054654, A190782 .

c[n_, k_] := Product[n-i, {i, 0, k-1}]/k!; row[n_] := CoefficientList[ n!*Sum[c[x, k], {k, 0, n}], x] // Reverse; Table[ row[n], {n, 0, 9}] // Flatten  (* Jean-François Alcover, Oct 04 2012 *)

T(n, k) = Sum_{i=0..k} Stirling1(i+n-k,n-k)*n!/(i+n-k)!. - Igor Victorovich Statsenko, May 27 2024

Missing 0 corrected by Steve Marak - N. J. A. Sloane, Jul 27 2012

A190782 Triangle T(n,k), read by rows, of the coefficients of x^k in the expansion of Sum_(m=0..n) binomial(x,m) = (a(k)*x^k)/n!, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 5, 0, 1, 24, 14, 11, -2, 1, 120, 94, 5, 25, -5, 1, 720, 444, 304, -75, 55, -9, 1, 5040, 3828, 364, 1099, -350, 112, -14, 1, 40320, 25584, 15980, -4340, 3969, -1064, 210, -20, 1

Offset: 0

Mokhtar Mohamed, Dec 29 2012

There is a strong relation between this triangle and triangle A048994 which deals with the binomial (x,n), this triangle being dealing with the summation of this binomial.

Apparently A054651 with reversed rows. - Mathew Englander, May 17 2014

			Triangle begins:
n\k     0       1       2       3       4       5       6      7     8
0       1
1       1       1
2       2       1       1
3       6       5       0        1
4      24      14      11       -2      1
5     120      94       5       25     -5       1
6     720     444     304      -75     55      -9       1
7    5040    3828     364     1099   -350     112     -14      1
8   40320   25584   15980    -4340   3969   -1064     210    -20     1
...

Seiichi Manyama, Rows n = 0..139, flattened

T(2*n,n) gives A347987.
Column 0-5 give A000142, A024167, A348063, A348064, A348065, A348068.
Cf. A048994, A054651, A132393.

row[n_] := CoefficientList[ Series[ Sum[ Binomial[x, m], {m, 0, n}], {x, 0, n}], x]*n!; Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 04 2013 *)

T(n,k) = T(n-1,k)+ T(n-1,k-1)- T(n-2,k-1)*(n-1)+ T(n-2,k)*(n-1)^2, T(n,n)=1, T(n,0)= n! for n >= 0.
T(n,k) = T(n-1,k)*n + (A048994(n,k)), T(n,n)= 1, T(n,0)= n! for n>= 0.
E.g.f. of column k: (log(1 + x))^k/(k! * (1 - x)). - Seiichi Manyama, Sep 26 2021
T(n, k) = Sum_{i=0..n-k} Stirling1(i+k, k)*n!/(i+k)!. - Igor Victorovich Statsenko, May 27 2024

A348063 Coefficient of x^2 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, 0, 11, 5, 304, 364, 15980, 34236, 1368936, 4429656, 173699712, 771653376, 30605906304, 175622947200, 7149130156800, 50800930272000, 2137822335475200, 18241636315507200, 796397873127782400, 7971407298921830400, 361615771356450508800, 4168685961862906982400, 196587429737202833817600

Offset: 2

Seiichi Manyama, Sep 26 2021

nonn

Column k=2 of A190782.

Cf. A000399, A028339, A054651, A347987, A348064, A348065, A348068.

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

a(n) = if(n<2, 0, a(n-1)+(n-1)^2*a(n-2)+(-1)^n*(n-2)!);

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^2/(2*(1-x))))

from sympy.abc import x
from sympy import ff, expand
def A348063(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**2) for k in range(2,n+1)) # Chai Wah Wu, Sep 27 2021

a(n) = a(n-1) + (n-1)^2 * a(n-2) + (-1)^n * (n-2)!.
E.g.f.: (log(1 + x))^2/(2 * (1 - x)).
a(n) ~ n! * log(2)^2 / 2 * (1 + (-1)^n*log(n)/(log(2)^2*n)). - Vaclav Kotesovec, Sep 27 2021

A348064 Coefficient of x^3 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, -2, 25, -75, 1099, -4340, 79064, -382060, 8550916, -48306984, 1303568760, -8346754416, 266955481584, -1894529909376, 70785236377728, -547468189825536, 23610353987137536, -196402650598402560, 9679304091074250240, -85687212859582878720, 4785340778000524477440

Offset: 3

Seiichi Manyama, Sep 26 2021

sign

Column k=3 of A190782.

Cf. A000399, A000454, A054651, A081051, A028340, A347987, A348063, A348065, A348068.

a(n) = n!*polcoef(sum(k=3, n, binomial(x, k)), 3);

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^3/(6*(1-x))))

from sympy.abc import x
from sympy import ff, expand
def A348064(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**3) for k in range(3,n+1)) # Chai Wah Wu, Sep 27 2021

E.g.f.: (log(1 + x))^3/(6 * (1 - x)).

A348065 Coefficient of x^4 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, -5, 55, -350, 3969, -31563, 408050, -3920950, 58206676, -657328100, 11111159696, -144321864960, 2747845864464, -40364369180016, 856755330487200, -14042902728462624, 329258021171239296, -5956512800554963584, 153050034289602269952, -3028534064042216488704, 84691080748928315003904

Offset: 4

Seiichi Manyama, Sep 26 2021

sign

Column k=4 of A190782.

Cf. A000482, A054651, A028341, A347987, A348063, A348064, A348068.

a(n) = n!*polcoef(sum(k=4, n, binomial(x, k)), 4);

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^4/(24*(1-x))))

from sympy.abc import x
from sympy import ff, expand
def A348065(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**4) for k in range(4,n+1)) # Chai Wah Wu, Sep 27 2021

E.g.f.: (log(1 + x))^4/(24 * (1 - x)).

A348068 Coefficient of x^5 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, -9, 112, -1064, 12873, -140595, 1870385, -23551110, 351042406, -5043110072, 84074954600, -1361614072000, 25218570009424, -455365645674480, 9298765013106384, -185409487083100320, 4144212593899945056, -90492302454898284864, 2199399908894486591040

Offset: 5

Seiichi Manyama, Sep 27 2021

sign

Column k=5 of A190782.

Cf. A000482, A001233, A054651, A347987, A348063, A348064, A348065.

a(n) = n!*polcoef(sum(k=5, n, binomial(x, k)), 5);

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^5/(120*(1-x))))

from sympy.abc import x
from sympy import ff, expand
def A348068(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**5) for k in range(5,n+1)) # Chai Wah Wu, Sep 27 2021

E.g.f.: (log(1 + x))^5/(120 * (1 - x)).

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)

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A054651 Triangle T(n,k) read by rows giving coefficients in expansion of n! * Sum_{i=0..n} C(x,i) in descending powers of x.

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A190782 Triangle T(n,k), read by rows, of the coefficients of x^k in the expansion of Sum_(m=0..n) binomial(x,m) = (a(k)*x^k)/n!, n >= 0, 0 <= k <= n.

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A348063 Coefficient of x^2 in expansion of n!* Sum_{k=0..n} binomial(x,k).

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A348064 Coefficient of x^3 in expansion of n!* Sum_{k=0..n} binomial(x,k).

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A348065 Coefficient of x^4 in expansion of n!* Sum_{k=0..n} binomial(x,k).

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A348068 Coefficient of x^5 in expansion of n!* Sum_{k=0..n} binomial(x,k).

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

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PARI

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