A141411 -id:A141411 - cp's OEIS frontend

A130620 Defined in comments.

3, 9, 31, 106, 365, 1263, 4388, 15336, 53871, 190059, 673222, 2393291, 8535397, 30526712, 109449848, 393272258, 1415768769, 5105086517, 18434398665, 66647658995, 241210652738, 873773659486, 3167642169823, 11491042716338, 41708741708554, 151461799255253

Offset: 0

Paul Curtz, Jun 18 2007

Given any sequence {u(i), i >= 0} we define a family of polynomials by P(0,x) = u(0), P(n,x) = u(n) + x*Sum_{i=0..n-1} u(i)*P(n-i-1, x).
Then a(n) is the sum of the odd coefficients of P(n,x) if n is odd and a(n) is the sum of the even coefficients otherwise: a(n) = ((-1)^n*P(n,-1) + P(n,1))/2.
For the present example we take {u(i)} to be 3,1,4,1,5,9,... (A000796).

			We have P(0,x)=3, P(1,x)=1+9x, P(2,x)=4+6x+27x^2, ..., so that for example a(2) = (25+37)/2 = 31.
The polynomials P(n,x) are:
n=0: 3,
n=1: 1+ 9*x,
n=2: 4+ 6*x+ 27*x^2,
n=3: 1+25*x+ 27*x^2+ 81*x^3,
n=4: 5+14*x+117*x^2+108*x^3+243*x^4,
n=5: 9+48*x+100*x^2+486*x^3+405*x^4+729*x^5.

P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.
P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78 .

Alois P. Heinz, Table of n, a(n) for n = 0..500

See A141411 for another version.

u:= proc(n) Digits:= max(n+10);
       trunc(10* frac(evalf(Pi*10^(n-1))))
    end:
P:= proc(n) option remember; local i, x;
      if n=0 then u(0)
    else unapply(expand(u(n)+x*add(u(i)*P(n-i-1)(x), i=0..n-1)), x)
      fi
    end:
a:= n-> (P(n)(1) +(-1)^n*P(n)(-1))/2:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 06 2009

nmax = 25; digits = RealDigits[Pi, 10, nmax+1][[1]]; p[0][] = digits[[1]]; p[n][x_] := p[n][x] = digits[[n+1]] + x*Sum[digits[[i+1]] p[n-i-1][x], {i, 0, n-1}]; a[n_] := (p[n][1] + (-1)^n*p[n][-1])/2; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 22 2012 *)

a(n) ~ c * d^n, where d = 3.6412947999106071671946396356753... (same as for A141411), c = 1.38770526630795733403509218... . - Vaclav Kotesovec, Sep 12 2014

Edited by N. J. A. Sloane, Aug 26 2009

Definition corrected and more terms from Alois P. Heinz, Sep 06 2009

A100597 Based on the first matrix inverse of transformed Bernoulli numbers as defined in the Comments line.

Original entry on oeis.org

1, 1, 2, 5, 14, 49, 258, 1385, 1342, -13739, 1727362, 20549165, -892047378, -13084315271, 979519187138, 16158974238545, -1747908612654946, -32246548780758179, 4903305033480792642, 100032668564662494485, -20685044415403212103730, -462550882810484735564351

Offset: 1

Paul Curtz, Jun 06 2007

sign

A family of polynomials is defined by P(0,x) = u(0), P(n,x) = u(n) +x*Sum_{i=0..n-1} u(i)*P(n-i-1,x), where u(n) is the n-th Bernoulli number. The coefficients of P(n-1,x) are used to fill the n-th row of the infinite lower triangle matrix M. Then a(n) is given by M^(-1)[n,1] * n!.

			a(3) = 2, because M = [1; -1/2 1; 1/6 -1 1; ...], M^(-1) = [1; 1/2 1; 1/3 1 1; ...], and (1/3)*3! = 2.

P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.
P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78.

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A027641/A027642, A130620, A141411.

P:= proc(n) option remember; local i, u, x; u:= bernoulli; `if`(n=0, u(0), unapply(expand(u(n) +x *add(u(i) *P(n-i-1)(x), i=0..n-1)), x)) end: a:= n-> (1/Matrix(n, (i, j)-> coeff(P(i-1)(x), x, j-1)))[n, 1] *n!: seq(a(n), n=1..30);  # Alois P. Heinz, Oct 12 2009

p[0, x_] = BernoulliB[0]; p[n_, x_] := p[n, x] = BernoulliB[n] + x*Sum[BernoulliB[i]*p[n-i-1, x], {i, 0, n-1}]; t[m_] := Table[ PadRight[CoefficientList[p[n, x], x], m+1], {n, 0, m}]; mmax = 20; Inverse[t[mmax-1]][[All, 1]]*Range[mmax]!
(* Jean-François Alcover, Jun 29 2011 *)

Edited and more terms from Alois P. Heinz, Oct 12 2009

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A130620 Defined in comments.

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