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.

Showing 1-3 of 3 results.

A105524 Diagonal sums of number triangle A105522.

Original entry on oeis.org

1, -2, 2, 0, -1, -2, 4, 4, -10, -12, 30, 36, -93, -114, 300, 372, -994, -1244, 3364, 4240, -11578, -14676, 40400, 51448, -142592, -182288, 508166, 651756, -1826037, -2348562, 6608844, 8520564, -24069258, -31097388, 88145436, 114096096, -324391422, -420590652, 1199074584
Offset: 0

Views

Author

Paul Barry, Apr 13 2005

Keywords

Programs

  • Mathematica
    CoefficientList[Series[((1+2*x)*Sqrt[1+4*x^2]-4*x^2-2*x-1)/(x^2*(Sqrt[1+4*x^2]-3)), {x,0,38}], x] (* Georg Fischer, Apr 09 2020 *)

Formula

G.f.: ((1+2*x)*sqrt(1+4*x^2)-4*x^2-2*x-1)/(x^2*(sqrt(1+4*x^2)-3)). - amended by Georg Fischer, Apr 09 2020
Conjecture: 2*(n+2)*(13*n^2-35*n+6)*a(n) +24*(n-8)*a(n-1) +(91*n^3-375*n^2+152*n+372)*a(n-2) +12*(8-n)*a(n-3) -4*(n-3)*(13*n^2-9*n-16)*a(n-4) = 0. - R. J. Mathar, Nov 09 2012

A105523 Expansion of 1-x*c(-x^2) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, -1, 0, 1, 0, -2, 0, 5, 0, -14, 0, 42, 0, -132, 0, 429, 0, -1430, 0, 4862, 0, -16796, 0, 58786, 0, -208012, 0, 742900, 0, -2674440, 0, 9694845, 0, -35357670, 0, 129644790, 0, -477638700, 0, 1767263190, 0
Offset: 0

Views

Author

Paul Barry, Apr 11 2005

Keywords

Comments

Row sums of A105522. Row sums of inverse of A105438.
First column of number triangle A106180.

Examples

			G.f. = 1 - x + x^3 - 2*x^5 + 5*x^7 - 14*x^9 + 42*x^11 - 132*x^13 + 429*x^15 + ...

Links

Crossrefs

Cf. A000108, A097331, A090192, A090181, A105522, A105438.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 + 2*x - Sqrt(1+4*x^2))/(2*x))); // G. C. Greubel, Sep 16 2018
  • Maple
    A105523_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w]:=-a[w-1]+(-1)^w*add(a[j]*a[w-j-1],j=1..w-1) od; convert(a,list)end: A105523_list(40); # Peter Luschny, May 19 2011
  • Mathematica
    a[n_?EvenQ] := 0; a[n_?OddQ] := 4^n*Gamma[n/2] / (Gamma[-n/2]*(n+1)!); a[0] = 1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
CoefficientList[Series[(1 + 2 x - Sqrt[1 + 4 x^2])/(2 x), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 01 2014 *)
a[ n_] := SeriesCoefficient[ (1 + 2 x - Sqrt[ 1 + 4 x^2]) / (2 x), {x, 0, n}]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], a[n] = -2 a[n - 1] + Sum[ a[j] a[n - j - 1], {j, 0, n - 1}]]; (* Michael Somos, Jun 17 2015 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = -2 * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */
  • Sage
    def A105523(n):
    if is_even(n): return 0 if n>0 else 1
    return -(sqrt(pi)*2^(n-1))/(gamma(1-n/2)*gamma((n+3)/2))
[A105523(n) for n in (0..29)] # Peter Luschny, Oct 31 2014

Formula

G.f.: (1 + 2*x - sqrt(1+4*x^2))/(2*x).
a(n) = 0^n + sin(Pi*(n-2)/2)(C((n-1)/2)(1-(-1)^n)/2).
G.f.: 1/(1+x/(1-x/(1+x/(1-x/(1+x/(1-x.... (continued fraction). - Paul Barry, Jan 15 2009
a(n) = Sum{k = 0..n} A090181(n,k)*(-1)^k. - Philippe Deléham, Feb 02 2009
a(n) = (1/n)*sum_{i = 0..n-1} (-2)^i*binomial(n, i)*binomial(2*n-i-2, n-1). - Vladimir Kruchinin, Dec 26 2010
With offset 1, a(n) = -2 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k), for n>1. - Michael Somos, Jul 25 2011
D-finite with recurrence: (n+3)*a(n+2) = -4*n*a(n), a(0)=1, a(1)=-1. - Fung Lam, Mar 18 2014
For nonzero terms, a(n) ~ (-1)^((n+1)/2)/sqrt(2*Pi)*2^(n+1)/(n+1)^(3/2). - Fung Lam, Mar 17 2014
a(n) = -(sqrt(Pi)*2^(n-1))/(Gamma(1-n/2)*Gamma((n+3)/2)) for n odd. - Peter Luschny, Oct 31 2014
From Peter Bala, Apr 20 2024: (Start)
a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n + k, 2*k)*Catalan(k), where Catalan(k) = A000108(k).
a(n) = (-2)^n * hypergeom([-n, n+1], [2], 1/2).
O.g.f.: A(x) = 1/x * series reversion of x*(1 - x)/(1 - 2*x). Cf. A152681. (End)

Extensions

Typo in definition corrected by Robert Israel, Oct 31 2014

A105438 Triangle, row sums = (Fibonacci numbers - 2).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 6, 5, 2, 1, 6, 9, 8, 6, 2, 1, 7, 12, 14, 10, 7, 2, 1, 8, 16, 20, 20, 12, 8, 2, 1, 9, 20, 30, 30, 27, 14, 9, 2, 1, 10, 25, 40, 50, 42, 35, 16, 10, 2, 1, 11, 30, 55, 70, 77, 56, 44, 18, 11, 2, 1
Offset: 0

Views

Author

Gary W. Adamson, Apr 09 2005

Keywords

Comments

Row sums = 1, 3, 6, 11, 19, 32, 53...(Fibonacci numbers - 2; starting with F(4)) The first few rows of the triangle are:
Row sums = (Fibonacci numbers - 2; starting 1, 3, 6...).
Column 1 = A002620; Column 2 = A006918; Column 3 = A096338.
Inverse array is A105522. - Paul Barry, Apr 11 2005
Diagonal sums are A027383(n). - Philippe Deléham, Jan 16 2014

Examples

			Column 2: 1, 2, 5, 8, 14, 20, 30...is generated by using the partial sum operator on 1, 1, 3, 3, 6, 6, 10, 10...
The first few rows of the triangle are:
  1;
  2, 1;
  3, 2, 1;
  4, 4, 2, 1;
  5, 6, 5, 2, 1;
  6, 9, 8, 6, 2, 1;
  7, 12, 14, 10, 7, 2, 1;
  8, 16, 20, 20, 12, 8, 2, 1;
  9, 20, 30, 30, 27, 14, 9, 2, 1;
  10, 25, 40, 50, 42, 35, 16, 10, 2, 1;
  ...

Links

Crossrefs

Cf. A001911, A002620, A006918, A096338, A105108.

Formula

By columns (k = 0, 1, 2...); use partial sum operator on (bin(n, k) numbers repeated).
T(n,k) = Sum_{j=0..n-k} C((j+2k)/2, k)*(1+(-1)^j)+C((j-1+2k)/2, k)*(1-(-1)^j)/2; Riordan array (1/(1-x)^2, x/(1-x^2)). - Paul Barry, Apr 11 2005
T(n,k) = T(n-1,k)+T(n-1,k-1)+T(n-2,k)-T(n-2,k-1)-T(n-3,k), T(0,0)=1, T(1,0)=2, T(1,1)= 1, T(n,k)= 0 if k<0 or if k>n. - Philippe Deléham, Jan 16 2014
Showing 1-3 of 3 results.

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