A064063 -id:A064063 - cp's OEIS frontend

A110518 Riordan array (1, x*c(3x)), c(x) the g.f. of A000108.

1, 0, 1, 0, 3, 1, 0, 18, 6, 1, 0, 135, 45, 9, 1, 0, 1134, 378, 81, 12, 1, 0, 10206, 3402, 756, 126, 15, 1, 0, 96228, 32076, 7290, 1296, 180, 18, 1, 0, 938223, 312741, 72171, 13365, 2025, 243, 21, 1, 0, 9382230, 3127410, 729729, 138996, 22275, 2970, 315, 24, 1, 0

Offset: 0

Paul Barry, Jul 24 2005

Row sums are C(3;n), A064063. Inverse is A110517. Diagonal sums are A110525.

			Rows begin
  1;
  0,    1;
  0,    3,    1;
  0,   18,    6,    1;
  0,  135,   45,    9,    1;
  0, 1134,  378,   81,   12,    1;
  ...
Production matrix begins:
  0,   1;
  0,   3,   1;
  0,   9,   3,   1;
  0,  27,   9,   3,   1;
  0,  81,  27,   9,   3,   1;
  0, 243,  81,  27,   9,   3,   1;
  ... - _Philippe Deléham_, Sep 23 2014

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0] := 1; T[0, k_] := 0; T[n_, k_] := (k/n)*3^(n - k)*Binomial[2*n - k - 1, n - k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)

concat([1], for(n=1,10, for(k=0,n, print1((k/n)*3^(n-k)*binomial(2*n-k-1,n-k), ", ")))) \\ G. C. Greubel, Aug 29 2017

Number triangle: T(0,k) = 0^k, T(n,k) = (k/n)*C(2n-k-1, n-k)*3^(n-k), n > 0, k > 0.
T(n,k) = A106566(n,k)*3^(n-k). - Philippe Deléham, Nov 08 2007
Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 3, 3, 3, 3, 3, 3, 3, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 23 2014

A366014 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 2 * A(x)).

Original entry on oeis.org

0, 1, 6, 54, 580, 6873, 86688, 1141500, 15512220, 215928900, 3063184410, 44124882750, 643692232404, 9490176205006, 141184118174640, 2116751269990968, 31951313566227228, 485159929343783532, 7405637373574690968, 113572576254948487800, 1749075343256441443320

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for pentagonal pyramidal numbers (with signs).

Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002293, A002411, A064063, A365754, A366015, A366016, A366017.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 2 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 n, n - k - 1] 2^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 2^k for n > 0.

A064311 Generalized Catalan numbers C(-2; n).

Original entry on oeis.org

1, 1, -1, 5, -25, 141, -849, 5349, -34825, 232445, -1582081, 10938709, -76616249, 542472685, -3876400305, 27919883205, -202480492905, 1477306676445, -10836099051105, 79861379898165, -591082795606425

Offset: 0

Wolfdieter Lang, Sep 21 2001

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Generalized Catalan numbers C(m; n): A000012 (m = 0), A000108 (m = 1), A064062 (m = 2), A064063 (m = 3), A064087 - A064093 (m = 4 thru 10); A064310 (m = -1) and A064325 - A064333 (m = -3 thru -11).

a[n_] := If[n==0, 1, Sum[(n-m)*Binomial[n+m-1, m]*(-2)^m/n, {m,0,n-1}]];
Table[a[n], {n,0,20}] (* Jean-François Alcover, Jun 03 2019 *)

import mpmath
mp.dps = 25; mp.pretty = True
a = lambda n: mpmath.hyp2f1(1-n, n, -n, -2) if n>0 else 1
[int(a(n)) for n in range(21)] # Peter Luschny, Nov 30 2014

a(n) = (1/n) * Sum_{m = 0..n-1} (n-m)*binomial(n-1+m, m)*(-2)^m = ((1/3)^n)*(1 + 2*Sum_{k = 0..n-1} C(k)*(-2*3)^k), for n >= 1, with a(0) := 1, and where C(n) = A000108(n), the Catalan numbers.
G.f.: (1+2*x*c(-2*x)/3)/(1-x/3) = 1/(1-x*c(-2*x)) with c(x) the g.f. of the Catalan numbers A000108.
a(n) = hypergeom([1-n, n], [-n], -2) for n>0. - Peter Luschny, Nov 30 2014
a(n) ~ -(-1)^n * 2^(3*n+1) / (25 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 03 2019
G.f. A(x) = 1 + series_reversion(x*(1 - (m-1)*x)/(1 + x)^2) at m = -2. - Peter Bala, Sep 08 2024

A323206 A(n, k) = hypergeometric([-k, k+1], [-k-1], n), square array read by ascending antidiagonals for n,k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 14, 1, 1, 5, 25, 67, 42, 1, 1, 6, 41, 190, 381, 132, 1, 1, 7, 61, 413, 1606, 2307, 429, 1, 1, 8, 85, 766, 4641, 14506, 14589, 1430, 1, 1, 9, 113, 1279, 10746, 55797, 137089, 95235, 4862, 1

Offset: 0

Peter Luschny, Feb 21 2019

Conjecture: A(n, k) is odd if and only if n is even or (n is odd and k + 2 = 2^j for some j > 0).

			Array starts:
    [n\k 0  1    2     3       4        5         6           7  ...]
    [0]  1, 1,   1,    1,      1,       1,        1,          1, ... A000012
    [1]  1, 2,   5,   14,     42,     132,      429,       1430, ... A000108
    [2]  1, 3,  13,   67,    381,    2307,    14589,      95235, ... A064062
    [3]  1, 4,  25,  190,   1606,   14506,   137089,    1338790, ... A064063
    [4]  1, 5,  41,  413,   4641,   55797,   702297,    9137549, ... A064087
    [5]  1, 6,  61,  766,  10746,  161376,  2537781,   41260086, ... A064088
    [6]  1, 7,  85, 1279,  21517,  387607,  7312789,  142648495, ... A064089
    [7]  1, 8, 113, 1982,  38886,  817062, 17981769,  409186310, ... A064090
    [8]  1, 9, 145, 2905,  65121, 1563561, 39322929, 1022586105, ... A064091
         A001844 A064096 A064302  A064303   A064304   A064305  diag: A323209
.
Seen as a triangle (by reading ascending antidiagonals):
                               1
                              1, 1
                            1, 2, 1
                           1, 3, 5, 1
                        1, 4, 13, 14, 1
                      1, 5, 25, 67, 42, 1
                   1, 6, 41, 190, 381, 132, 1

J. Abate, W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011).
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687.

Rows: A000108, A064062, A064063, A064087, A064088, A064089, A064090, A064091.
Columns: A001844, A064096, A064302, A064303, A064304, A064305.
Diagonals: A323209 (main), A323208 (sup main), A323217 (sub main).
Sums of antidiagonals: A323207
Cf. A064094, A009766, A238762.

# The function ballot is defined in A238762.
A := (n, k) -> add(ballot(2*j, 2*k)*n^j, j=0..k):
for n from 0 to 6 do seq(A(n, k), k=0..9) od;
# Or by recurrence:
A := proc(n, k) option remember;
if n = 1 then return `if`(k = 0, 1, (4*k + 2)*A(1, k-1)/(k + 2)) fi:
if k < 2 then return [1, n+1][k+1] fi; n*(4*k - 2);
((%*(n - 1) - k - 1)*A(n, k-1) + %*A(n, k-2))/((n - 1)*(k + 1)) end:
for n from 0 to 6 do seq(A(n, k), k=0..9) od;
# Alternative:
Arow := proc(n, len) # Function REVERT is in Sloane's 'Transforms'.
[seq(1 + n*k, k=0..len-1)]; REVERT(%); seq((-1)^k*%[k+1], k=0..len-1) end:
for n from 0 to 8 do Arow(n, 8) od;

A[n_, k_] := Hypergeometric2F1[-k, k + 1, -k - 1, n];
Table[A[n, k], {n, 0, 8}, {k, 0, 8}]
(* Alternative: *)
prev[f_, n_] := InverseSeries[Series[-x f, {x, 0, n}]]/(-x);
f[n_, x_] := (1 + (n - 1) x)/((1 - x)^2);
For[n = 0, n < 9, n++, Print[CoefficientList[prev[f[n, x], 8], x]]]
(* Continued fraction: *)
num[k_, n_] := If[k < 2, 1, If[k == 2, -x, -n x]];
cf[n_, len_] := ContinuedFractionK[num[k, n], 1, {k, len + 2}];
Arow[n_, len_] := Rest[CoefficientList[Series[cf[n, len], {x, 0, len}], x]];
For[n = 0, n < 9, n++, Print[Arow[n, 8]]]

{A(n,k) = polcoeff((1/x)*serreverse(x*((1+(n-1)*(-x))/((1-(-x))^2)+x*O(x^k))), k)}
for(n=0, 8, for(k=0, 8, print1(A(n, k), ", ")); print())

# Valid for n > 0.
def genCatalan(n): return SR(1/(x- x^2*(1 - sqrt(1 - 4*x*n))/(2*x*n)))
for n in (1..8): print(genCatalan(n).series(x).list())
# Alternative:
def pseudo_reversion(g, invsign=false):
    if invsign: g = g.subs(x=-x)
    g = g.shift(1)
    g = g.reverse()
    g = g.shift(-1)
    return g
R. = PowerSeriesRing(ZZ)
for n in (0..6):
    f = (1+(n-1)*x)/((1-x)^2)
    s = pseudo_reversion(f, true)
    print(s.list())

A(n, k) = [x^k] 1/(x - x^2*C(n*x)) if n > 0 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the generating function of the Catalan numbers A000108.
A(n, k) = Sum_{j=0..k} (binomial(2*k-j, k) - binomial(2*k-j, k+1))*n^(k-j).
A(n, k) = Sum_{j=0..k} binomial(k + j, k)*(1 - j/(k + 1))*n^j (cf. A009766).
A(n, k) = 1 + Sum_{j=0..k-1} ((1+j)*binomial(2*k-j, k+1)/(k-j))*n^(k-j).
A(n, k) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^k)/(1+(n-1)*x), n>0.
A(n, k) ~ ((4*n)^k/(Pi^(1/2)*k^(3/2)))*(1+1/(2*n-1))^2.
If we shift the series f with constant term 1 to the right, invert it with respect to composition and shift the result back to the left then we call this the 'pseudo reversion' of f, prev(f). Row n of the array gives the coefficients of the pseudo reversion of f = (1 + (n - 1)*x)/((1 - x)^2) with an additional inversion of sign. Note that f is not revertible. See also the Sage implementation below.
A(n, k) = [x^k] prev((1 + (n - 1)*(-x))/(1 - (-x))^2).
A(n, k) = [x^(k+1)] cf(n, x) where cf(n, x) = K_{i>=1} c(i)/b(i) in the notation of Gauß with b(i) = 1, c(1) = 1, c(2) = -x and c(i) = -n*x for i > 2.
For a recurrence see the Maple section.

A116866 Generalized Catalan triangle of Riordan type, called C(1,3).

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 25, 25, 7, 1, 190, 190, 55, 10, 1, 1606, 1606, 472, 94, 13, 1, 14506, 14506, 4300, 898, 142, 16, 1, 137089, 137089, 40861, 8785, 1495, 199, 19, 1, 1338790, 1338790, 400567, 87826, 15655

Offset: 0

Wolfdieter Lang, Mar 24 2006

This triangle is the second 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(a(n,m)*x^n,m=0..n) is D(x,z)=g(z)/(1 - x*z*c(3*z))= g(z)*(3*z-x*z*(1-3*z*c(3*z)))/(3*z-x*z+(x*z)^2), with g(z) and c(z) defined below.
This is the Riordan triangle named (g(x),x*c(3*x)) with g(x):=(1+3*x*c(3*x)/2)/(1+x/2) and c(x) is the o.g.f. of A000108 (Catalan numbers). g(x) is the o.g.f. of A064063 (C(3;n) Catalan generalization).
For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.

			[1];[1,1];[4,4,1];[25,25,7,1];[190,190,55,10,1];...
Production matrix begins:
1, 1
3, 3, 1
9, 9, 3, 1
27, 27, 9, 3, 1
81, 81, 27, 9, 3, 1
243, 243, 81, 27, 9, 3, 1
... _Philippe Deléham_, Sep 22 2014

Wolfdieter Lang, First 10 rows.

Row sums give A116867.
Compare with the row reversed and scaled triangle A116868 (called Y(1, 3)).
Cf. A115193 (similar sequence C(1,2)).

G.f. for column m>=0 is g(x)*(x*c(3*x))^m, with g(x):=(1+3*x*c(3*x)/2)/(1+x/2) and c(x) is the o.g.f. of A000108 (Catalan numbers).

A365668 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 2 * A(x)).

Original entry on oeis.org

0, 1, 7, 73, 905, 12354, 179305, 2715192, 42414021, 678476755, 11058588574, 182999237590, 3066447596459, 51926183715280, 887204891847960, 15276037569668880, 264797324173666845, 4617195655522976361, 80930337327794271445, 1425171253004955494215, 25202145191953299213490

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for 4-dimensional figurate numbers A001296 (with signs).

Eric Weisstein's World of Mathematics, Series Reversion

Cf. A001296, A002294, A064063, A365755, A366014, A366035, A366036, A366037.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 2 x)/(1 + x)^5, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 2^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(5*n,n-k-1) * 2^k for n > 0.

a(n) ~ sqrt(32 - 19*sqrt(5/2)) * 3^(4*n - 3/2) * 5^(3*n) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 3/2) * (25 + 34*sqrt(10))^n). - Vaclav Kotesovec, Sep 27 2023

A385474 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+x)^3 ).

Original entry on oeis.org

1, 7, 76, 991, 14281, 219172, 3512440, 58096591, 984340003, 16996883887, 298017184048, 5291703108292, 94961611382860, 1719543577996888, 31379622840361696, 576519956457976495, 10655055147825932119, 197959348525977645781, 3695112941037246866044

Offset: 0

Seiichi Manyama, Aug 01 2025

nonn

Index entries for reversions of series

Cf. A064063, A385475.

Cf. A384950.

my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^2/(1+x)^3)/x)

a(n) = sum(k=0, n, 2^(n-k)*binomial(3*(n+1), k)*binomial(3*n-k+1, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(3*(n+1),k) * binomial(3*n-k+1,n-k).

a(n) = (1/(n+1)) * [x^n] ( (1+x)^3 / (1-2*x)^2 )^(n+1).

A157491 A050165*A130595 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 2, -6, 5, 0, -5, 20, -28, 14, 0, 14, -70, 135, -120, 42, 0, -42, 252, -616, 770, -495, 132, 0, 132, -924, 2730, -4368, 4004, -2002, 429, 0, -429, 3432, -11880, 23100, -27300, 19656, -8008, 1430

Offset: 0

Philippe Deléham, Mar 01 2009

Triangle, read by rows, given by [0,-1,-1,-1,-1,-1,-1,...] DELTA [1,1,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938. Triangle related to k-regular trees.

			Triangle begins:
  1;
  0,  1;
  0, -1,  2;
  0,  2, -6,   5;
  0, -5, 20, -28, 14;
  ...

Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A000108, A062991, A094385.

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.

Sum_{k=0..n} T(n,k)*x^(n-k) = A064093, A064092, A064091, A064090, A064089, A064088, A064087, A064063, A064062, A000108, A000012, A064310, A064311, A064325, A064326, A064327, A064328, A064329, A064330, A064331, A064332, A064333 for x = -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 respectively. [Philippe Deléham, Mar 03 2009]

A385475 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^3 / (1+x)^4 ).

Original entry on oeis.org

1, 10, 154, 2836, 57601, 1244584, 28063288, 652821724, 15551944804, 377503375150, 9303441938506, 232168129150420, 5854967533764766, 148981015820615968, 3820184959840942564, 98616983735455104412, 2560818171703792341484, 66845502538144505160040

Offset: 0

Seiichi Manyama, Aug 01 2025

nonn

Index entries for reversions of series

Cf. A064063, A385474.

Cf. A386723.

my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^3/(1+x)^4)/x)

a(n) = sum(k=0, n, 2^(n-k)*binomial(4*(n+1), k)*binomial(4*n-k+2, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(4*(n+1),k) * binomial(4*n-k+2,n-k).

a(n) = (1/(n+1)) * [x^n] ( (1+x)^4 / (1-2*x)^3 )^(n+1).

A115188 Second diagonal (M=2) of triangle A115154 (called Y(3,1)).

Original entry on oeis.org

1, 13, 115, 1036, 9688, 93571, 927523, 9387580, 96634294, 1008719002, 10653244294, 113630599852, 1222361061310, 13246563382735, 144479425979635, 1584810988780420, 17471975574010630, 193493968384827670

Offset: 0

Wolfdieter Lang, Feb 23 2006

			115 = a(2) = A064063(4) - 3* A064063(3) = 190 - 3*25.

a(n)= A115154(n+1,n+1), n>=0.
Recurrence: a(n)= b(n) - 3*b(n-1), with b(n):= A064063(n+2), n>=0.
G.f.: ((-3+2*x) + 3*(1-3*x)*c(3*x))/(x*(2+x)), with the o.g.f. c(x) of A000108 (Catalan).

cp's OEIS Frontend

A110518 Riordan array (1, x*c(3x)), c(x) the g.f. of A000108.

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A366014 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 2 * A(x)).

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Mathematica

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A064311 Generalized Catalan numbers C(-2; n).

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Sage

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A323206 A(n, k) = hypergeometric([-k, k+1], [-k-1], n), square array read by ascending antidiagonals for n,k >= 0.

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Maple

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A116866 Generalized Catalan triangle of Riordan type, called C(1,3).

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A365668 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 2 * A(x)).

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Mathematica

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A385474 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+x)^3 ).

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PARI

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A157491 A050165*A130595 as infinite lower triangular matrices.

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