A006632 -id:A006632 - cp's OEIS frontend

A329059 3-parking triangle T(r, i, 3) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 3 and 0 <= i <= r.

1, 3, 1, 15, 9, 3, 91, 78, 48, 16, 612, 680, 600, 375, 125, 4389, 5985, 6840, 6156, 3888, 1296, 32890, 53130, 74382, 86779, 79233, 50421, 16807, 254475, 475020, 786240, 1123200, 1331200, 1228800, 786432, 262144, 2017356, 4272048, 8155728, 13762791, 19978245, 23973894, 22320522, 14348907, 4782969

Offset: 0

Stefano Spezia, Nov 03 2019

The k-parking numbers interpolate between the generalized Fuss-Catalan numbers and the number of parking functions (see Yip).

			r/i|      0      1      2      3      4
———————————————————————————————————————
0  |      1
1  |      3      1
2  |     15      9      3
3  |     91     78     48     16
4  |    612    680    600    375    125
...

Stefano Spezia, First 151 rows of the triangle, flattened
Martha Yip, A Fuss-Catalan variation of the caracol flow polytope, arXiv:1910.10060 [math.CO], 2019.

Cf. A000108, A000272, A006632, A007318, A328978 (row sums), A329057, A329058, A329060.

T[r_, i_,k_] := (r + 1)^(i-1)*Binomial[k*(r + 1) + r - i - 1, r - i]; Flatten[Table[T[r,i,3],{r,0,8},{i,0,r}]]

T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i).
T(r, 0, 3) = A006632(r + 1).
T(r, r, 3) = A000272(r + 1).

A337291 a(n) = 3binomial(4n,n)/(4*n-1).

Original entry on oeis.org

4, 12, 60, 364, 2448, 17556, 131560, 1017900, 8069424, 65204656, 535070172, 4446927732, 37353738800, 316621743480, 2704784196240, 23263187479980, 201275443944432, 1750651680235920, 15298438066553776, 134252511729576240, 1182622941581590080

Offset: 1

Lucas A. Brown, Aug 21 2020

a(n) is the number of lattice paths from (0,0) to (3n,n) using only the steps (1,0) and (0,1) and whose only lattice points on the line y = x/3 are the path's endpoints. - Lucas A. Brown, Aug 21 2020

R. J. Mathar, The Eggenberger-Polya urn process: Probabilities of revisited ball ratios, vixra:2502 (2025)

Cf. A006632, A337292.

Array[3 Binomial[4 #, #]/(4 # - 1) &, 21] (* Michael De Vlieger, Aug 21 2020 *)

a(n) = {3*binomial(4*n,n)/(4*n-1)} \\ Andrew Howroyd, Aug 21 2020

a(n) = 4*A006632(n).
G.f.: 4*x*F(x)^3 where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
D-finite with recurrence 3*n*(3*n-1)*(3*n-2)*a(n) -8*(4*n-5)*(4*n-3)*(2*n-1)*a(n-1)=0, a(0)=1. - R. J. Mathar, Jan 26 2025

A369617 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^3 + x) ).

Original entry on oeis.org

1, 4, 22, 146, 1079, 8525, 70468, 601816, 5268241, 47019566, 426250277, 3914020148, 36328457669, 340278596273, 3212416054283, 30534649412247, 291981031204917, 2806832429353512, 27109863184695640, 262951127248539898, 2560229132085602215

Offset: 0

Seiichi Manyama, Jan 27 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..984
Index entries for reversions of series

Cf. A007317, A369616, A370844.

Cf. A006632, A274735.

A369617 := proc(n)
    add(binomial(n+1,k) * binomial(4*n-4*k+2,n-k),k=0..n) ;
    %/(n+1) ;
end proc;
seq(A369617(n),n=0..70) ; # R. J. Mathar, Jan 28 2024

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^3+x))/x)

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

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

D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-91*n^3 -32*n^2 +n+2)*a(n-1) +2*(n-1)*(465*n^2 -337*n+86)*a(n-2) -4*(n-1)*(n-2) *(219*n-187)*a(n-3) +283*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 28 2024

A371495 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1+x))^3.

Original entry on oeis.org

1, 3, 12, 64, 381, 2430, 16227, 112008, 792717, 5721165, 41945373, 311529831, 2338909219, 17722127580, 135346614906, 1040779011412, 8051611785006, 62620659604659, 489339248275242, 3840135625895886, 30251386980891657, 239138782521553659, 1896380840948325606

Offset: 0

Seiichi Manyama, Mar 25 2024

nonn

Cf. A001006, A371494, A371496.

Cf. A006632.

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

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

A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 9, 22, 1, 4, 15, 52, 140, 1, 5, 22, 91, 340, 969, 1, 6, 30, 140, 612, 2394, 7084, 1, 7, 39, 200, 969, 4389, 17710, 53820, 1, 8, 49, 272, 1425, 7084, 32890, 135720, 420732, 1, 9, 60, 357, 1995, 10626, 53820, 254475, 1068012, 3362260

Offset: 0

Henry Bottomley, Mar 12 2002

For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).

Antidiagonals of convolution matrix of Table 1.5, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019

			Rows start
  1;
  1,   1;
  1,   2,   4;
  1,   3,   9,  22;
  1,   4,  15,  52, 140;
etc.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

Columns include A000012, A000027, A055999.
Right-hand diagonals include A002293, A069271, A006632.
Cf. A130458 (row sums).

A069270 := proc(n,k)
        binomial(n+3*k,k)*(n-k+1)/(n+2*k+1) ;
end proc: # R. J. Mathar, Oct 11 2015

Table[Binomial[n + 3 k, k] (n - k + 1)/(n + 2 k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 27 2019 *)

T(n, k) = C(n+3k, k)*(n-k+1)/(n+2k+1).
For n >= k+3: T(n, k) = T(n-2, k+1)-T(n-3, k+1).
T(n, n) = T(n+2, n-1) = C(4n, n)/(3n+1).

A247031 G.f.: 1 = Sum_{n>=0} a(n)x^n [ Sum_{k=0..n+1} binomial(n+1, k)^3 * (-x)^k ]^3.

Original entry on oeis.org

1, 3, 69, 5005, 806148, 239220375, 116532061510, 86173621173099, 91417549409916684, 133300597778263476112, 258360728839130761571757, 647880493609691058921741273, 2055869510173976408422116133220, 8103111707775918586405906798540650, 39047811321420953231675462397758519802

Offset: 0

Paul D. Hanna, Sep 09 2014

nonn

Compare to a g.f. of A006632(n) = 3*binomial(4*n+3,n)/(4*n+3):
1 = Sum_{n>=0} A006632(n)*x^n * [ Sum_{k=0..n+1} binomial(n+1, k)*(-x)^k ]^3
where A006632 equals the self-convolution cube of A002293.
...
a(3*n) == 2 (mod 3) iff A245658(n) == 2 (mod 3), where A245658 is the self-convolution cube root of this sequence (conjecture).

			O.g.f.: A(x) = 1 + 3*x + 69*x^2 + 5005*x^3 + 806148*x^4 + 239220375*x^5 +...
such that the coefficients satisfy:
1 = 1*(1-x)^3 + 3*x*(1-2^3*x+x^2)^3 + 69*x^2*(1-3^3*x+3^3*x^2-x^3)^3 + 5005*x^3*(1-4^3*x+6^3*x^2-4^3*x^3+x^4)^3 + 806148*x^4*(1-5^3*x+10^3*x^2-10^3*x^3+5^3*x^4-x^5)^3 +...
Note that the cube-root of the o.g.f., A(x)^(1/3), is an integer series:
A(x)^(1/3) = 1 + x + 22*x^2 + 1624*x^3 + 264962*x^4 + 79136637*x^5 + 38671111558*x^6 + 28642761340956*x^7 + 30413158977739302*x^8 +...+ A245658(n)*x^n +...
CONJECTURE: given
G(x,m,j) = Sum_{n>=0} a(n) * (m*x)^n * [ Sum_{k=0..n+1} C(n+1, k)^3 * (j*x)^k ]^3
then G(x,m,j)^(1/3) is an integer series in x whenever m, j, are integers.
OBSERVATIONS.
The terms of this sequence are congruent to 2 modulo 3 at positions:
[12, 36, 93, 108, 111, 114, 120, 174, 255, 279, ...].
The terms of A245658 are congruent to 2 modulo 3 at positions:
[4, 12, 31, 36, 37, 38, 40, 58, 85, 93, ...].

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A245658, A247030, A212370.

{a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, m+1, binomial(m+1, k)^3 * (-x)^k +x*O(x^n) )^3 ), n))}
for(n=0, 20, print1(a(n), ", "))

A366327 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^3).

Original entry on oeis.org

1, 2, -5, 33, -260, 2263, -20979, 203124, -2030121, 20786694, -216928144, 2298911699, -24673591005, 267644087524, -2929602893537, 32317666058508, -358931896710948, 4010200327457883, -45040693394259858, 508253687784232108, -5759468659295939684

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366326, A366328.

Cf. A006632, A364407.

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

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

A377504 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^3.

Original entry on oeis.org

1, 3, 36, 735, 21972, 871995, 43308378, 2588123811, 180990517032, 14507325973395, 1311719669172750, 132102208441613883, 14666354372331521676, 1779817542971018697003, 234399632982398657764578, 33297612755940733707395955, 5075234637265322738651060688, 826215756199826873368252279971

Offset: 0

Seiichi Manyama, Oct 30 2024

nonn

Cf. A295238, A377503, A377528.

Cf. A006632, A364987.

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

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364987.

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

A381745 Expansion of exp( Sum_{k>=1} binomial(8k-1,2k) * x^k/k ).

Original entry on oeis.org

1, 21, 903, 49525, 3070308, 204928371, 14369906538, 1043861319189, 77866470852108, 5929621690613108, 459076176165983247, 36026517938705145267, 2859318461620989381900, 229114879928544260792946, 18509862380800289696106372, 1506048000721264678984095445, 123303480420582227597300406588

Offset: 0

Seiichi Manyama, Mar 05 2025

Seiichi Manyama, Table of n, a(n) for n = 0..514

Cf. A079489, A381744, A381746.

Cf. A006632, A381751.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(8*k-1, 2*k)*x^k/k)))

G.f. A(x) satisfies A(x^2) = B(x)/x * B(-x)/(-x), where B(x) is the g.f. of A006632.
a(n) = Sum_{k=0..2*n} (-1)^k * A006632(k+1) * A006632(2*n-k+1).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(8*k-1,2*k) * a(n-k).
G.f.: B(x)^3, where B(x) is the g.f. of A381751.

A365123 G.f. satisfies A(x) = (1 + x / (1 - x*A(x))^4)^2.

Original entry on oeis.org

1, 2, 9, 44, 244, 1438, 8858, 56340, 367160, 2438934, 16453015, 112411836, 776258588, 5409237100, 37988571802, 268606426836, 1910584687932, 13661702623498, 98148312810335, 708092115326436, 5127976641997944, 37264674894021280, 271650189521574734

Offset: 0

Seiichi Manyama, Aug 22 2023

nonn

Cf. A006632, A365114, A365124.

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

If g.f. satisfies A(x) = (1 + x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*(n-k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).

cp's OEIS Frontend

A329059 3-parking triangle T(r, i, 3) read by rows: T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i) with k = 3 and 0 <= i <= r.

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A337291 a(n) = 3*binomial(4*n,n)/(4*n-1).

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A369617 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^3 + x) ).

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A371495 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1+x))^3.

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A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

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A247031 G.f.: 1 = Sum_{n>=0} a(n)*x^n * [ Sum_{k=0..n+1} binomial(n+1, k)^3 * (-x)^k ]^3.

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A366327 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^3).

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A377504 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^3.

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A329059 3-parking triangle T(r, i, 3) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 3 and 0 <= i <= r.

A337291 a(n) = 3binomial(4n,n)/(4*n-1).

A247031 G.f.: 1 = Sum_{n>=0} a(n)x^n [ Sum_{k=0..n+1} binomial(n+1, k)^3 * (-x)^k ]^3.

A381745 Expansion of exp( Sum_{k>=1} binomial(8k-1,2k) * x^k/k ).