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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A075513 Triangle read by rows. T(n, m) are the coefficients of Sidi polynomials.

Original entry on oeis.org

1, -1, 2, 1, -8, 9, -1, 24, -81, 64, 1, -64, 486, -1024, 625, -1, 160, -2430, 10240, -15625, 7776, 1, -384, 10935, -81920, 234375, -279936, 117649, -1, 896, -45927, 573440, -2734375, 5878656, -5764801, 2097152, 1, -2048, 183708, -3670016, 27343750, -94058496, 161414428, -134217728, 43046721
Offset: 1

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

Coefficients of the Sidi polynomials (-1)^(n-1)*D_{n-1,1,n-1}(x), for n >=1, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980].
The row polynomials p(n, x) := Sum_{m=0..n-1} a(n, m)x^m, n >= 1, are obtained from ((Eu(x)^n)*(x-1)^n)/(n*x), where Eu(x) := xd/dx is the Euler-derivative with respect to x.
The row polynomials p(n, y) := Sum_{m=0..n-1} a(n, m)*y^m, n >= 1, are also obtained from ((d^m/dx^m)((exp(x)-1)^m)/m)/exp(x) after replacement of exp(x) by y. Here (d^m/dx^m)f(x), m >= 1, denotes m-fold differentiation of f(x) with respect to x.
b(k,m,n) := (Sum_{p=0..m-1} (a(m, p)*((p+1)*k)^n))/(m-1)!, n >= 0, has g.f. 1/Product_{p=1..m} (1 - k*p*x) for k = 1, 2,... and m = 1, 2,...
The (signed) row sums give A000142(n-1), n >= 1, (factorials) and (unsigned) A074932(n).
The (unsigned) columns give A000012 (powers of 1), 2*A001787(n+1), (3^2)*A027472(n), (4^3)*A038846(n-1), (5^4)*A036071(n-5), (6^5)*A036084(n-6), (7^6)*A036226(n-7), (8^7)*A053107(n-8) for m=0..7.
Right edge of triangle is A000169. - Michel Marcus, May 17 2013

Examples

			The triangle T(n, m)  begins:
  n\m 0     1      2        3        4         5         6          7       8
  1:  1
  2: -1     2
  3:  1    -8      9
  4: -1    24    -81       64
  5:  1   -64    486    -1024      625
  6: -1   160  -2430    10240   -15625      7776
  7:  1  -384  10935   -81920   234375   -279936    117649
  8: -1   896 -45927   573440 -2734375   5878656  -5764801    2097152
  9:  1 -2048 183708 -3670016 27343750 -94058496 161414428 -134217728 4304672
  [Reformatted by _Wolfdieter Lang_, Oct 12 2022]
-----------------------------------------------------------------------------
p(2,x) = -1+2*x = (1/(2*x))*x*(d/dx)*x*(d/dx)*(x-1)^2.

References

  • A. Sidi, Practical Extrapolation Methods: Theory and Applications, Cambridge University Press, Cambridge, 2003.

Links

Crossrefs

Cf. A075510, A075511, A075512, A074932, A075515, A075516, A075906..A075925, A076002..A076013, A258773.

Programs

  • Maple
    # Assuming offset 0.
seq(seq((-1)^(n-k)*binomial(n, k)*(k+1)^n, k=0..n), n=0..8);
# Alternative:
egf := x -> 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y):
ser := x -> series(egf(x), x, 12):
row := n -> seq(coeff(n!*coeff(ser(x), x, n), y, k), k=0..n):
seq(print(row(n)), n = 0..8); # Peter Luschny, Oct 21 2022
  • Mathematica
    p[n_, x_] := p[n, x] = Nest[ x*D[#, x]& , (x-1)^n, n]/(n*x); a[n_, m_] := Coefficient[ p[n, x], x, m]; Table[a[n, m], {n, 1, 9}, {m, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 03 2013 *)
  • PARI
    tabl(nn) = {for (n=1, nn, for (m=0, n-1, print1((-1)^(n-m-1)*binomial(n-1, m)*(m+1)^(n-1), ", ");); print(););} \\ Michel Marcus, May 17 2013

Formula

T(n, m) = ((-1)^(n-m-1)) binomial(n-1, m)*(m+1)^(n-1), n >= m+1 >= 1, else 0.
G.f. for m-th column: ((m+1)^m)(x/(1+(m+1)*x))^(m+1), m >= 0.
E.g.f.: -LambertW(-x*y*exp(-x))/((1+LambertW(-x*y*exp(-x)))*x*y). - Vladeta Jovovic, Feb 13 2008 [corrected for offset 0 <= m <= n. For offset n >= 1 take the integral over x. - Wolfdieter Lang, Oct 12 2022]
T(n, k) = S(n, k+1) / n where S(, ) is triangle in A258773. - Michael Somos, May 13 2018
E.g.f. of column k, with offset n >= 0: exp(-(k + 1)*x)*((k + 1)*x)^k/k!. - Wolfdieter Lang, Oct 20 2022
E.g.f: 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y) assuming offset = 0. - Peter Luschny, Oct 21 2022

A201595 E.g.f. satisfies A(x) = exp(x*A(x)) * cosh(x*A(x)).

Original entry on oeis.org

1, 1, 4, 28, 288, 3936, 67328, 1385728, 33372160, 921118720, 28677169152, 994360565760, 38007586684928, 1587878686621696, 71990467473965056, 3520403893852831744, 184707311409882464256, 10350444842488122310656, 616975843658373414256640, 38981881007475178476666880
Offset: 0

Views

Author

Paul D. Hanna, Dec 03 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! +...
The coefficients of x^n/n! in initial powers of G(x) = (1 + exp(2*x))/2 begin:
G^1: [(1), 1, 2, 4, 8, 16, 32, 64, 128, ...];
G^2: [1,(2), 6, 20, 72, 272, 1056, 4160, ...];
G^3: [1, 3,(12), 54, 264, 1368, 7392, 41184, ...];
G^4: [1, 4, 20,(112), 680, 4384, 29600, 207232, ...];
G^5: [1, 5, 30, 200,(1440), 11000, 88080, 732800, ...];
G^6: [1, 6, 42, 324, 2688,(23616), 217392, 2080224, ...];
G^7: [1, 7, 56, 490, 4592, 45472,(471296), 5076400, ...];
G^8: [1, 8, 72, 704, 7344, 80768, 928512,(11085824), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 112/4, 1440/5, 23616/6, 471296/7, 11085824/8, ...].

Links

Crossrefs

Cf. A214225, A201594, A074932, A218300, A218301, A218302.
Cf. A370907, A370908.

Programs

  • Mathematica
    Join[{1},Table[Sum[Binomial[n+1,k] k^n/(n+1),{k,0,n+1}]/2,{n,20}]] (* Harvey P. Dale, Feb 04 2012 *)
CoefficientList[Series[(x-LambertW[-x*E^x])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 04 2012 *)
  • PARI
    a(n)=n!*polcoeff(1/x*serreverse(x*exp(-x+x^2*O(x^n))/cosh(x+x^2*O(x^n))),n)
  • PARI
    a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp((n+1)*X)*cosh(X)^(n+1)/(n+1),n)
  • PARI
    a(n)=sum(k=0,n+1,binomial(n+1,k)*k^n/(n+1)/2)
  • PARI
    /* Formula for a(n,m) where A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!: */
{a(n,m=1)=sum(k=0, n+m, binomial(n+m, k)*k^n*m/(n+m)/2^m)}
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(A=sum(k=0,n,(k+1)^(k-1)*cosh((k+1)*x+x*O(x^n))*x^k/k!));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Oct 24 2012
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(A=1+sum(k=1,n,k^k*sinh(k*x+x^2*O(x^n))/(k*x)*x^k/k!));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Nov 20 2012

Formula

a(n) = (1/2) * Sum_{k=0..n+1} C(n+1,k) * k^n / (n+1).
a(n) = [x^n/n!] exp((n+1)*x) * cosh(x)^(n+1) / (n+1).
E.g.f. A(x) satisfies:
(1) A( x*exp(-x)/cosh(x) ) = exp(x)*cosh(x).
(2) A(x) = (1/x)*Series_Reversion( x*exp(-x)/cosh(x) ).
(3) A(x) = (1 + exp(2*x*A(x)))/2.
(4) A(x) = exp(G(x)) where G(x) is the e.g.f. of A074932.
(5) A(x) = Sum_{n>=0} (n+1)^(n-1) * cosh((n+1)*x) * x^n/n!. - Paul D. Hanna, Oct 24 2012
(6) A(x) = 1 + Sum_{n>=1} n^n * sinh(n*x)/(n*x) * x^n/n!. - Paul D. Hanna, Nov 20 2012
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = Sum_{k=0..n+m} C(n+m, k) * k^n * m/(n+m) / 2^m.
a(n) = A214225(n+1)/(n+1).
E.g.f.: (x-LambertW(-x*exp(x)))/(2*x). - Vaclav Kotesovec, Dec 04 2012
a(n) ~ n!*sqrt(LambertW(exp(-1))+1)/(2*sqrt(2*Pi)*n^(3/2)*LambertW(exp(-1))^(n+1)). - Vaclav Kotesovec, Dec 04 2012
G.f.: 1/2 + 1/2 * Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1). - Seiichi Manyama, Apr 23 2024
a(n) = n! * Sum_{k=0..n} 2^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024
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