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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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A098118 a(n) = n!*[x^n] (log(x+1) * Sum_{j=0..n} C(2*n,j)*x^j).

Original entry on oeis.org

1, 7, 74, 1066, 19524, 434568, 11393808, 343976400, 11752855200, 448372820160, 18892607771520, 871406506494720, 43669963405555200, 2362804077652300800, 137275789612950374400, 8523776656311156172800, 563309040416875548364800
Offset: 1

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Author

Alexander Adamchuk, Oct 25 2004

Keywords

Comments

Previous name was: Sum of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n) multiplied by (2*n-1)!/n!.
Let A(i, j) denote an infinite array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i times to the harmonic sequence. For example, the first row starts as 1, 5/2, 13/3, ..., and the next row begins with 1, 7/2, 47/6, and so forth. Then a(n) equals n!*A(n, n) for all n. - John M. Campbell, Jan 20 2019

Examples

			n=2: HilbertMatrix[n,n]
  1 1/2
  1/2 1/3
so a(2) = (2*2-1)! / 2! * (1 + 1/2 + 1/2 + 1/3) = 7.
The n X n Hilbert matrix begins:
  1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
  1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
  1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
  1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
  1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
G.f. = x + 7*x^2 + 74*x^3 + 1066*x^4 + 19524*x^5 + 434568*x^6 + ...

Links

Crossrefs

Cf. A000984, A001008, A002805, A005249, A078791, A008275, A086881.
Cf. A257635 (column 1).

Programs

  • Maple
    A098118 := n -> n!*coeff(series(ln(x+1)*add(binomial(2*n,j)*x^j, j=0..n), x, n+1), x, n): seq(A098118(n),n=1..17); # Peter Luschny, Jan 18 2015
A098118 := n -> hypergeom([1,1,1-n],[2,n+2],1)*n*(2*n)!/(n+1)!:
seq(simplify(A098118(n)), n=1..17); # Peter Luschny, Jun 11 2016
A098118 := n -> sum(abs(Stirling1(n,k))*k*(n+1)^(k-1), k=1..n):
seq(A098118(n), n=1..17); # Ondrej Kutal, Oct 20 2021
  • Mathematica
    Table[(2n - 1)!/n! Sum[ 1/(i + j - 1), {i, n}, {j, n}], {n, 17}]
a[ n_] := If[ n < 1, 0, (2 n)! / n! Sum[ -(-1)^k / k, {k, 2 n}]]; (* Michael Somos, Dec 09 2013 *)
a[ n_] := If[ n < 1, 0, (2 n - 1)! / n! Sum[ 1 / (i + j - 1), {i, n}, {j, n}]]; (* Michael Somos, Apr 14 2015 *)
a[ n_] := If[ n < 1, 0, n! SeriesCoefficient[ (Log[ EllipticNomeQ[ m] / (m/16)]) EllipticK[ m] 16^n / (Binomial[2 n, n] 2 Pi), {m, 0, n}]]; (* Michael Somos, Apr 14 2015 *)
a[ n_] := If[ n < 1, 0, (2 n + 1)! / n! SeriesCoefficient[ PolyLog[2, -1] + PolyLog[2, (1 - x)/2] + Log[(1 + x)/2] Log[(1 - x)/2]/2 + Log[(1 + x)/(1 - x)] Log[2]/2, {x, 0, 2 n + 1}]]; (* Michael Somos, Apr 14 2015 *)
  • PARI
    {a(n) = if( n<1, 0, (2*n)! / n! * sum( k=1, 2*n, -(-1)^k / k))}; /* Michael Somos, Dec 09 2013 */

Formula

a(n) = ((2*n-1)!/n!)*Sum_{i=1..n} Sum_{j=1..n} 1/(i+j-1).
a(n) = 2*(2*n-1)!/(n-1)!*H'(2*n), where H'(2*n) = H(2*n) - H(n), H'(n) = Sum_{k=1..n} (1/k)*(-1)^(k+1) is an alternate signs Harmonic number, H(n) = Sum_{k=1..n} 1/k is a Harmonic number, H(n) = A001008/A002805. - Alexander Adamchuk, Oct 25 2004
Sum_{k>0} a(k) * k! * x^(2*k + 1) / (2*k + 1)! = F(-1) + F((1 - x)/2) + log(2) * log((1 + x) / (1 - x)) / 2 + log((1 + x) / 2) * log((1 - x) / 2) / 2 where F(x) = Li_2(x) is the dilogarithm function. - Michael Somos, Dec 09 2013
2 * A078791(n) = a(n) * A000984(n). - Michael Somos, Apr 14 2015
a(n) = (2*n)!/n! * Sum_{k = 1..n} 1/(n + k). Column 1 of A257635. - Peter Bala, Nov 05 2015
E.g.f.: (log((sqrt(1-4*x)+1)/2)*(-3*x+sqrt(1-4*x)*(x-1)+1))/(4*x^2+sqrt(1-4*x)*(3*x-1)-5*x+1). - Vladimir Kruchinin, Jun 04 2016
a(n) = hypergeom([1,1,1-n], [2,n+2], 1)*n*(2*n)!/(n+1)!. - Peter Luschny, Jun 11 2016
a(n) ~ log(2) * 2^(2*n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Jul 10 2016
a(n) = Sum_{k=1..n} |s(n,k)|*k*(n+1)^(k-1) where s(n,k) are Stirling numbers of the first kind (A008275). - Ondrej Kutal, Oct 20 2021
a(n) = n! * [x^n] (-log(1 - x)/(1 - x)^(n+1)). - Seiichi Manyama, May 20 2025

Extensions

New name from Peter Luschny, Jan 19 2015

A349727 Triangle read by rows, T(n, k) = [x^(n - k)] hypergeom([-n, -1 + n], [-n], x).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 4, 3, 2, 1, 15, 10, 6, 3, 1, 56, 35, 20, 10, 4, 1, 210, 126, 70, 35, 15, 5, 1, 792, 462, 252, 126, 56, 21, 6, 1, 3003, 1716, 924, 462, 210, 84, 28, 7, 1, 11440, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1, 43758, 24310, 12870, 6435, 3003, 1287, 495, 165, 45, 9, 1
Offset: 0

Views

Author

Peter Luschny, Nov 27 2021

Keywords

Examples

			Triangle starts:
[0] 1;
[1] 0,     1;
[2] 1,     1,    1;
[3] 4,     3,    2,    1;
[4] 15,    10,   6,    3,    1;
[5] 56,    35,   20,   10,   4,   1;
[6] 210,   126,  70,   35,   15,  5,   1;
[7] 792,   462,  252,  126,  56,  21,  6,   1;
[8] 3003,  1716, 924,  462,  210, 84,  28,  7,  1;
[9] 11440, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1;

Crossrefs

Row sums: A088218, alternating row sums: A091526.
Central coefficients: binomial(3*n-2, n) (cf. A117671).
T(n, 0) = binomial(2*(n-1), n) (cf. A001791).
Cf. A257635.

Programs

  • Maple
    p := n -> hypergeom([-n, -1 + n], [-n], x):
seq(seq(coeff(simplify(p(n)), x, n - k), k = 0..n), n = 0..10);
  • Mathematica
    (* rows[0..k], k[0..oo] *)
r={};k=11;For[n=0,nDetlef Meya, Jun 26 2023 *)
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