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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A025564 Triangular array, read by rows: pairwise sums of trinomial array A027907.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 4, 3, 1, 1, 4, 8, 10, 8, 4, 1, 1, 5, 13, 22, 26, 22, 13, 5, 1, 1, 6, 19, 40, 61, 70, 61, 40, 19, 6, 1, 1, 7, 26, 65, 120, 171, 192, 171, 120, 65, 26, 7, 1, 1, 8, 34, 98, 211, 356, 483, 534, 483, 356, 211, 98, 34, 8, 1, 1, 9, 43, 140, 343, 665, 1050, 1373
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Counting the top row as row 0, T(n,k) is the number of strings of nonnegative integers "s(1)s(2)s(3)...s(k)" such that s(1)+s(2)+s(3)+...+s(k) = n and the string does not contain the substring "00". E.g., T(3,5) = 8 because the valid strings are 02010, 01020, 11010, 10110, 10101, 01110, 01101 and 01011. T(4,3) = 13, counting 040, 311, 301, 130, 031, 103, 013, 220, 202, 022, 211, 121 and 112. - Jose Luis Arregui (arregui(AT)unizar.es), Dec 05 2007

Examples

			                  1
              1   2   1
          1   3   4   3   1
      1   4   8  10   8   4   1
  1   5  13  22  26  22  13   5   1

Crossrefs

Columns include A025565, A025566, A025567, A025568.
Cf. A025177.

Programs

  • Mathematica
    T[, 0] = 1; T[1, 1] = 2; T[n, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Jul 22 2018 *)
  • PARI
    {T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,2,1][k+1], if( n==2, [1,3,4,3,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};
  • PARI
    T(n,k)=polcoeff(Ser(polcoeff(Ser((1+y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)
  • PARI
    {T(n, k) = if( k<0 || k>2*n, 0, if(n==0, 1, polcoeff( (1 + x + x^2)^n, k)+ polcoeff( (1 + x + x^2)^(n-1), k-1)))};

Formula

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 2, 1], [1, 3, 4, 3, 1].
G.f.: (1+yz)/[1-z(1+y+y^2)].

Extensions

Edited by Ralf Stephan, Jan 09 2005
Edited by Clark Kimberling, Jun 20 2012

A025178 First differences of the central trinomial coefficients A002426.

Original entry on oeis.org

0, 2, 4, 12, 32, 90, 252, 714, 2032, 5814, 16700, 48136, 139152, 403286, 1171380, 3409020, 9938304, 29017878, 84844044, 248382516, 727971360, 2135784798, 6272092596, 18435108258, 54228499920, 159636389850, 470256930052, 1386170197704
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Previous name was: "a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0 = s(n), |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n), where T is the array defined in A025177."
Note that n-1 divides a(n) for n>=2. - T. D. Noe, Mar 16 2005

Links

Crossrefs

Cf. A001006, A002426, A005717, A007971, A025177, A079309, A097893, A105696, A162551.

Programs

  • Maple
    a := n -> 2*(n-1)*hypergeom([1-n/2, 3/2-n/2], [2], 4):
seq(simplify(a(n)), n=1..28); # Peter Luschny, Oct 29 2015
  • Mathematica
    Rest[Differences[CoefficientList[Series[x/Sqrt[1-2x-3x^2],{x,0,30}],x]]] (* Harvey P. Dale, Aug 22 2011 *)
Differences[Table[Hypergeometric2F1[(1-n)/2,1-n/2,1,4],{n,1,29}]] (* Peter Luschny, Nov 03 2015 *)
  • PARI
    a(n) = sum(k=1, n\2, binomial(n-1,2*k-1)*binomial(2*k,k)); \\ Altug Alkan, Oct 29 2015
  • Sage
    def a():
    b, c, n = 0, 2, 2
    yield b
    while True:
        yield c
        b, c = c, ((2*n-1)*c+3*(n-1)*b)*n//((n+1)*(n-1))
        n += 1
A025178 = a()
print([next(A025178) for  in (1..20)]) # _Peter Luschny, Nov 04 2015

Formula

a(n) = T(n,n) for n>=1, where T is the array defined in A025177.
a(n) = A002426(n+1) - A002426(n). - Benoit Cloitre, Nov 02 2002
a(n) is asymptotic to c*3^n/sqrt(n) with c around 1.02... - Benoit Cloitre, Nov 02 2002
a(n) = 2*(n-1)*A001006(n-2). - M. F. Hasler, Oct 24 2011
a(n) = 2*A005717(n-1). - R. J. Mathar, Jul 09 2012
E.g.f. Integral(Integral(2*exp(x)*((1-1/x)*BesselI(1,2*x) + 2*BesselI(0,2*x)))). - Sergei N. Gladkovskii, Aug 16 2012
G.f.: -1/x + (1/x-1)/sqrt(1-2*x-3*x^2). - Sergei N. Gladkovskii, Aug 16 2012
D-finite with recurrence: a(n) = ((2+n)*a(n-2)+3*(3-n)*a(n-3)+3*(n-1)*a(n-1))/n, a(0)=1, a(1)=0, a(2)=2. - Sergei N. Gladkovskii, Aug 16 2012 [adapted to new offset by Peter Luschny, Nov 04 2015]
G.f.: (1-x)/x^2*G(0) - 1/x^2 , where G(k)= 1 + x*(2+3*x)*(4*k+1)/( 4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
From Peter Bala, Oct 28 2015: (Start)
a(n) = Sum_{k = 0..floor(n/2)} binomial(n-1,2*k-1)*binomial(2*k,k). Cf. A097893.
n*(n-2)*a(n) = (2*n-3)*(n-1)*a(n-1) + 3*(n-1)*(n-2)*a(n-2) with a(1) = 0, a(2) = 2. (End)
From Peter Luschny, Oct 29 2015: (Start)
a(n) = 2*(n-1)*hypergeom([1-n/2,3/2-n/2],[2],4).
a(n) = (n-1)!*[x^(n-1)](2*exp(x)*BesselI(1,2*x)).
a(n) = (n-1)*A007971(n) for n>=2.
A105696(n) = a(n-1) + a(n) for n>=2.
A162551(n-2) = (1/2)*Sum_{k=1..n} binomial(n,k)*a(k) for n>=2.
A079309(n) = (1/2)*Sum_{k=1..2*n} (-1)^k*binomial(2*n,k)*a(k) for n>=1.
(End)

Extensions

New name based on a comment by T. D. Noe, Mar 16 2005, offset set to 1 and a(1) = 0 prepended by Peter Luschny, Nov 04 2015
Previous Showing 21-22 of 22 results.

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