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 1-6 of 6 results.

A167435 Hankel transform of A084076.

Original entry on oeis.org

1, 2, -16, -3584, -1114112, -771751936, -68719476736, 50102545854496768, 4999067643975288487936, 1104958199127771065681444864, 363815722265501838229553819942912
Offset: 0

Views

Author

Paul Barry, Nov 03 2009

Keywords

Links

Programs

  • Mathematica
    a[n_] := Sum[Binomial[n + k, 2*k]*(-2)^(n - k), {k, 0, n}]; Table[2^(n^2)*(-1)^n*a[n], {n, 0, 10}] (* G. C. Greubel, Jun 13 2016 *)

Formula

a(n) = 2^(n^2)*(-1)^n*Sum_{k=0..n} C(n+k,2k)*(-2)^(n-k).
a(n) = 2^(n^2)*[x^n](1-2*x)/(1+3*x+4*x^2) = 2^(n^2) *A247560(n).

A115193 Generalized Catalan triangle of Riordan type, called C(1,2).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 13, 13, 5, 1, 67, 67, 27, 7, 1, 381, 381, 157, 45, 9, 1, 2307, 2307, 963, 291, 67, 11, 1, 14589, 14589, 6141, 1917, 477, 93, 13, 1, 95235, 95235, 40323, 12867, 3363, 723, 123, 15, 1, 636925
Offset: 0

Views

Author

Wolfdieter Lang, Feb 23 2006

Keywords

Comments

This triangle is the first of a family of generalizations of the Catalan convolution triangle A033184 (which belongs to the Bell subgroup of the Riordan group).
The o.g.f. of the row polynomials P(n,x):=Sum_{m=0..n} a(n,m)*x^n is D(x,z) = g(z)/(1 - x*z*c(2*z)) = g(z)*(2*z-x*z*(1-2*z*c(2*z)))/(2*z-x*z+(x*z)^2), with g(z) and c(z) defined below.
This is the Riordan triangle named (g(x),x*c(2*x)) with g(x):=(1+2*x*c(2*x))/(1+x) and c(x) is the o.g.f. of A000108 (Catalan numbers). g(x) is the o.g.f. of A064062 (C(2;n) Catalan generalization).
The column sequences (without leading zeros) are A064062, A064062(n+1), A084076, A115194, A115202-A115204, for m=0..6.
For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.

Examples

			Triangle begins:
   1;
   1,  1;
   3,  3,  1;
  13, 13,  5,  1;
  67, 67, 27,  7,  1;
  ...
Production matrix begins:
    1,   1;
    2,   2,   1;
    4,   4,   2,   1;
    8,   8,   4,   2,   1;
   16,  16,   8,   4,   2,   1;
   32,  32,  16,   8,   4,   2,   1;
   64,  64,  32,  16,   8,   4,   2,   1;
  128, 128,  64,  32,  16,   8,   4,   2,   1;
  ... _Philippe Deléham_, Sep 22 2014

Links

Crossrefs

Row sums give A115197. Compare with the row reversed and scaled triangle A115195.
Cf. A116866 (similar sequence C(1,3)).

Programs

  • Maple
    lim:=7: c:=(1-sqrt(1-8*x))/(4*x): g:=(1+2*x*c)/(1+x): gf1:=g*(x*c)^m: for m from 0 to lim do t:=taylor(gf1,x,lim+1): for n from 0 to lim do a[n,m]:=coeff(t,x,n):od:od: seq(seq(a[n,m],m=0..n),n=0..lim); # Nathaniel Johnston, Apr 30 2011
  • Mathematica
    A110510[n_, k_] := (k/n)*Binomial[2*n - k - 1, n - k]*2^(n - k);
T[n_, k_] := If[n == 0, 1, Sum[A110510[n, i], {i, k, n}]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 17 2025 *)

Formula

G.f. for column m>=0 is g(x)*(x*c(2*x))^m, with g(x):=(1+2*x*c(2*x))/(1+x) and c(x) is the o.g.f. of A000108 (Catalan numbers).
T(n,k) = Sum_{i=k..n} A110510(n,i) for 0 <= k <= n. - Werner Schulte, Mar 24 2019

A115195 Triangle of numbers, called Y(1,2), related to generalized Catalan numbers A062992(n) = C(2;n+1) = A064062(n+1).

Original entry on oeis.org

1, 2, 3, 4, 10, 13, 8, 28, 54, 67, 16, 72, 180, 314, 381, 32, 176, 536, 1164, 1926, 2307, 64, 416, 1488, 3816, 7668, 12282, 14589, 128, 960, 3936, 11568, 26904, 51468, 80646, 95235, 256, 2176, 10048, 33184, 86992, 189928, 351220, 541690, 636925, 512, 4864
Offset: 0

Views

Author

Wolfdieter Lang, Feb 23 2006

Keywords

Comments

This triangle Y(1,2) appears in the totally asymmetric exclusion process for the (unphysical) values alpha=1, beta=2. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.
The main diagonal (M=1) gives the generalized Catalan sequence C(2,n+1):=A064062(n+1).
The diagonal sequences give A064062(n+1), 2*A084076, 4*A115194, 8*A115202, 16*A115203, 32*A115204 for n+1>= M=1,..,6.

Examples

			Triangle begins:
   1;
   2,  3;
   4, 10,  13;
   8, 28,  54,  67;
  16, 72, 180, 314, 381;
  ...

Links

Crossrefs

Row sums give A084076.

Formula

G.f. m-th diagonal, m>=1: ((1 + 2*x*c(2*x))*(2*x*c(2*x))^m)/(2*x*(1+x)) with c(x) the o.g.f. of A000108 (Catalan).

A084075 Length of list created by n substitutions k -> Range( -abs(k+1), abs(k-1), 2) starting with {1}.

Original entry on oeis.org

1, 2, 5, 12, 33, 86, 249, 680, 2033, 5722, 17485, 50260, 156033, 455534, 1431281, 4228752, 13412193, 40003058, 127840085, 384232156, 1235575201, 3737280582, 12080678505, 36736735672, 119276490193, 364372758986, 1187542872989
Offset: 0

Views

Author

Wouter Meeussen, May 11 2003

Keywords

Examples

			{1}, {-2,0}, {-1,1,3,-1,1}, {0,2,-2,0,-4,-2,0,2,0,2,-2,0}

Links

Crossrefs

Cf. A084076, A084077, A084078.
Cf. A027307, A215067, A034015 (even bisection).

Programs

  • Magma
    I:=[1,2,5,12]; [n le 4 select I[n] else (6*(35*n^2-55*n-76)*Self(n-1) + (275*n^4-770*n^3-203*n^2+1736*n-912)*Self(n-2) -6*(5*n^2+5*n-28)*Self(n-3) + (n-4)*(n-2)*(25*n^2+5*n-48)*Self(n-4))/(n*(n+2)*(25*n^2-45*n-28)): n in [1..41]]; // G. C. Greubel, Nov 24 2022
  • Mathematica
    Rest@CoefficientList[InverseSeries[Series[ (-1-6n-8n^2+(1+2n)^2 Sqrt[1+4n])/( 2(n+4n^2+4n^3)), {n, 0, 40}]], n]
Length/@Flatten/@NestList[ #/.k_Integer:>Range[-Abs[k+1], Abs[k-1], 2] &, {1}, 8]
  • Python
    # replace iterates lists as described in Example.
def replace(L):
    return [i for k in L for i in range(-abs(k + 1), 1 + abs(k - 1), 2)]
def a(n):
  L = [1]
  for k in range(n): L=replace(L)
  return len(L)
print([a(n) for n in range(12)]) # F. Chapoton, Nov 15 2024
  • SageMath
    @CachedFunction
def a(n): # a = A084075
    if n < 4: return (1, 2, 5, 12)[n]
    else: return (6*(35*n^2 +15*n -96)*a(n-1) +(275*n^4+330*n^3-863*n^2+120*n+126)*a(n-2) -6*(5*n^2+15*n-18)*a(n-3) +(n-3)*(n-1)*(25*n^2+55*n-18)*a(n-4))/((n+1)*(n+3)*(25*n^2+5*n-48))
[a(n) for n in range(41)] # G. C. Greubel, Nov 24 2022

Formula

G.f. is the series reversion of (-1 -6*x -8*x^2 + (1+2*x)^2 * sqrt(1+4*x))/(2*(x +4*x^2 +4*x^3)).
a(2*n) = A027307(n)/2, n >= 1.
a(n) = ( 6*(35*n^2 +15*n -96)*a(n-1) + (275*n^4 +330*n^3 -863*n^2 +120*n +126)*a(n-2) - 6*(5*n^2 +15*n -18)*a(n-3) + (n-3)*(n-1)*(25*n^2 +55*n -18)*a(n-4) )/((n+1)*(n+3)*(25*n^2 +5*n -48)), n >= 4. - G. C. Greubel, Nov 24 2022

A084077 Length of list created by n substitutions k -> Range(-abs(k+1), abs(k-1)) starting with {1}.

Original entry on oeis.org

1, 3, 11, 41, 159, 633, 2575, 10657, 44735, 190017, 815231, 3527681, 15378687, 67478401, 297777407, 1320753665, 5884652543, 26326301697, 118211192831, 532574203905, 2406726828031, 10906541371393
Offset: 0

Views

Author

Wouter Meeussen, May 11 2003

Keywords

Examples

			{1}, {-2,-1,0}, {-1,0,1,2,3,0,1,2,-1,0,1}

Links

Crossrefs

Cf. A084075, A084076, A084078.

Programs

  • Magma
    I:=[1,3,11]; [n le 3 select I[n] else (3*(7*n^2 -11*n +6)*Self(n-1) + 2*(28*n^2 -51*n +14)*Self(n-2) + 4*(n-2)*(7*n-4)*Self(n-3))/((n+2)*(7*n-11)): n in [1..41]]; // G. C. Greubel, Nov 23 2022
  • Mathematica
    Length/@Flatten/@NestList[ # /. k_Integer:>Range[ -Abs[k+1], Abs[k-1]]&, {1}, 8]
Flatten[{1,RecurrenceTable[{(n+3)*(7*n-4)*a[n] == 3*(7*n^2+3*n+2)*a[n-1] + 2*(28*n^2+5*n-9)*a[n-2] + 4*(n-1)*(7*n+3)*a[n-3],a[1]==3,a[2]==11,a[3]==41},a,{n,20}]}] (* Vaclav Kotesovec, Oct 14 2012 *)
  • SageMath
    @CachedFunction
def a(n):  # a = A084077
    if (n<3): return (1,3,11)[n]
    else: return (3*(7*n^2 +3*n +2)*a(n-1) + 2*(28*n^2 +5*n -9)*a(n-2) + 4*(n-1)*(7*n+3)*a(n-3))/((n+3)*(7*n-4))
[a(n) for n in range(31)] # G. C. Greubel, Nov 23 2022

Formula

invOGF satisfies n - (1+3*n)*a(n) - 2*n*(1+n)*a(n)^2 - 2*n^2*a(n)^3 = 0. [Is it true?]
Recurrence: (n+3)*(7*n-4)*a(n) = 3*(7*n^2+3*n+2)*a(n-1) + 2*(28*n^2+5*n-9)*a(n-2) + 4*(n-1)*(7*n+3)*a(n-3). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(52+34*sqrt(2))*(2+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

A167432 Riordan array (c(2x)^2,xc(2x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 4, 1, 20, 6, 1, 112, 36, 8, 1, 672, 224, 56, 10, 1, 4224, 1440, 384, 80, 12, 1, 27456, 9504, 2640, 600, 108, 14, 1, 183040, 64064, 18304, 4400, 880, 140, 16, 1, 1244672, 439296, 128128, 32032, 6864, 1232, 176, 18, 1, 8599552, 3055104, 905216, 232960
Offset: 0

Views

Author

Paul Barry, Nov 03 2009

Keywords

Comments

Inverse of (1-4x+4x^2,x(1-2x)) (A167431). Row sums are A084076. First column is A003645.

Examples

			Triangle begins
1,
4, 1,
20, 6, 1,
112, 36, 8, 1,
672, 224, 56, 10, 1,
4224, 1440, 384, 80, 12, 1,
27456, 9504, 2640, 600, 108, 14, 1,
183040, 64064, 18304, 4400, 880, 140, 16, 1,
1244672, 439296, 128128, 32032, 6864, 1232, 176, 18, 1,
8599552, 3055104, 905216, 232960, 52416, 10192, 1664, 216, 20, 1,
60196864, 21498880, 6449664, 1697280, 396032, 81536, 14560, 2184, 260, 22, 1
The production matrix is
4, 1,
4, 2, 1,
8, 4, 2, 1,
16, 8, 4, 2, 1,
32, 16, 8, 4, 2, 1,
64, 32, 16, 8, 4, 2, 1,
128, 64, 32, 16, 8, 4, 2, 1,
256, 128, 64, 32, 16, 8, 4, 2, 1,
512, 256, 128, 64, 32, 16, 8, 4, 2, 1
When topped with the row (1,0,0,0...), this has inverse
1,
-4, 1,
4, -2, 1,
0, 0, -2, 1,
0, 0, 0, -2, 1,
0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, 0, 0, 0, -2, 1

Formula

Number triangle T(n,k)=A054445(n,k)*2^(n-k).
Showing 1-6 of 6 results.

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