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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A118345 Pendular triangle, read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), else T(n,k) = T(n,n-1-k) + 2*T(n-1,k), for n>=k>=0, with T(n,0) = 1 and T(n,n) = 0^n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 5, 1, 0, 1, 4, 11, 6, 1, 0, 1, 5, 18, 30, 7, 1, 0, 1, 6, 26, 70, 40, 8, 1, 0, 1, 7, 35, 121, 201, 51, 9, 1, 0, 1, 8, 45, 184, 487, 286, 63, 10, 1, 0, 1, 9, 56, 260, 873, 1445, 386, 76, 11, 1, 0, 1, 10, 68, 350, 1375, 3592, 2147, 502, 90, 12, 1, 0
Offset: 0

Views

Author

Paul D. Hanna, Apr 26 2006

Keywords

Comments

See A118340 for definition of pendular triangles and pendular sums.

Examples

			Row 6 equals the pendular sums of row 5:
  [1,  4, 11,  6,  1,  0], where the pendular sums proceed as follows:
  [1, __, __, __, __, __]: T(6,0) = T(5,0) = 1;
  [1, __, __, __, __,  1]: T(6,5) = T(6,0) + 2*T(5,5) = 1 + 2*0 = 1;
  [1,  5, __, __, __,  1]: T(6,1) = T(6,5) + T(5,1) = 1 + 4 = 5;
  [1,  5, __, __,  7,  1]: T(6,4) = T(6,1) + 2*T(5,4) = 5 + 2*1 = 7;
  [1,  5, 18, __,  7,  1]: T(6,2) = T(6,4) + T(5,2) = 7 + 11 = 18;
  [1,  5, 18, 30,  7,  1]: T(6,3) = T(6,2) + 2*T(5,3) = 18 + 2*6 = 30;
  [1,  5, 18, 30,  7,  1, 0] finally, append a zero to obtain row 6.
Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  1,   0;
  1,  3,  5,   1,    0;
  1,  4, 11,   6,    1,    0;
  1,  5, 18,  30,    7,    1,    0;
  1,  6, 26,  70,   40,    8,    1,   0;
  1,  7, 35, 121,  201,   51,    9,   1,  0;
  1,  8, 45, 184,  487,  286,   63,  10,  1,  0;
  1,  9, 56, 260,  873, 1445,  386,  76, 11,  1, 0;
  1, 10, 68, 350, 1375, 3592, 2147, 502, 90, 12, 1, 0; ...
Central terms are T(2*n,n) = A118346(n);
semi-diagonals form successive self-convolutions of the central terms:
T(2*n+1,n) = A118347(n) = [A118346^2](n),
T(2*n+2,n) = A118348(n) = [A118346^3](n).

Links

Crossrefs

Cf. A118346, A118347, A118348, A118349, A118340.
Cf. A167763 (p=0), A118340 (p=1), this sequence (p=2), A118350 (p=3).

Programs

  • Magma
    function T(n,k,p)
  if k lt 0 or n lt k then return 0;
  elif k eq 0 then return 1;
  elif k eq n then return 0;
  elif n gt 2*k then return T(n,n-k,p) + T(n-1,k,p);
  else return T(n,n-k-1,p) + p*T(n-1,k,p);
  end if;
  return T;
end function;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021
  • Mathematica
    T[n_, k_, p_]:= T[n,k,p] = If[nG. C. Greubel, Feb 17 2021 *)
  • PARI
    T(n,k)=if(n2*k,T(n,n-k)+T(n-1,k),T(n,n-1-k)+2*T(n-1,k)))))
  • Sage
    @CachedFunction
def T(n, k, p):
    if (k<0 or n2*k): return T(n,n-k,p) + T(n-1,k,p)
    else: return T(n, n-k-1, p) + p*T(n-1, k, p)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

Formula

T(2*n+m,n) = [A118346^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A118346.

A118346 Central terms of pendular triangle A118345.

Original entry on oeis.org

1, 1, 5, 30, 201, 1445, 10900, 85128, 682505, 5585115, 46461437, 391743850, 3340361700, 28755475180, 249572076200, 2181469638880, 19186562661273, 169677521094215, 1507881643936015, 13458730170115778, 120599648894147185
Offset: 0

Views

Author

Paul D. Hanna, Apr 26 2006

Keywords

Comments

Also, g.f. A(x) = (1/x)*series_reversion of x/(1 + x*GF(A005572)), where GF(A005572) is the g.f. of A005572, which is the number of walks on cubic lattice starting and finishing on the xy plane and never going below it.

Links

Crossrefs

Cf. A118345, A118347, A118348, A118349.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
[1] cat Coefficients(R!( Reversion( x/((1+x)*(1+4*x+x^2)) ) )); // G. C. Greubel, Mar 17 2021
  • Mathematica
    CoefficientList[1 +InverseSeries[Series[x/((1+x)*(1+4*x+x^2)), {x,0,30}]], x] (* G. C. Greubel, Mar 17 2021 *)
  • PARI
    {a(n) = polcoeff(serreverse( x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^n)))/(2*(1-2*x)))/x, n)}
  • Sage
    def A118346_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ( x/((1+x)*(1+4*x+x^2)) ).reverse() ).list()
a=A118346_list(31); [1]+a[1:] # G. C. Greubel, Mar 17 2021

Formula

G.f.: A=A(x) satisfies A = 1 - 2*x*A + 2*x*A^2 + x*A^3.
G.f.: A(x) = 1 + series_reversion( x/((1+x)*(1+4*x+x^2)) ).
G.f.: A(x) = (1/x)*series_reversion( x*(1-2*x + sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) ).
For n>0: a(n) = (1/n)*Sum_{j=0..n} Sum_{i=0..n-1} ( binomial(n,j) * binomial(j,i) * binomial(n-j,2*j-n-i-1) * 5^(2*n-3*j+2*i+1) ). -Vladimir Kruchinin, Dec 26 2010

A118348 Semi-diagonal (two rows below central terms) of pendular triangle A118345 and equal to the self-convolution cube of the central terms (A118346).

Original entry on oeis.org

1, 3, 18, 121, 873, 6606, 51728, 415629, 3407391, 28388847, 239675406, 2045980440, 17629939980, 153142537440, 1339599358944, 11789960853293, 104327344928619, 927627432162129, 8283625668834238, 74259685465582569, 668054892245119353
Offset: 0

Views

Author

Paul D. Hanna, Apr 26 2006

Keywords

Links

Crossrefs

Cf. A118345, A118346, A118347, A118349.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (Reversion( x*(1-2*x +Sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) )/x)^3 )); // G. C. Greubel, Mar 17 2021
  • Mathematica
    CoefficientList[(InverseSeries[Series[x*(1-2*x +Sqrt[(1-2*x)*(1-6*x)])/(2*(1-2*x)), {x, 0, 30}]]/x)^3, x] (* G. C. Greubel, Mar 17 2021 *)
  • PARI
    {a(n) = polcoeff( (serreverse(x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^n)))/(2*(1-2*x)))/x)^3, n)}
  • Sage
    def A118347_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (( x*(1-2*x +sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) ).reverse()/x)^3 ).list()
A118347_list(31) # G. C. Greubel, Mar 17 2021

Formula

G.f.: ( series_inverse( x*(1-2*x +sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) )/x )^3.
Showing 1-3 of 3 results.

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