A329057 -id:A329057 - cp's OEIS frontend

A329096 Row sums of A329057.

1, 2, 8, 47, 374, 3852, 49398, 762785, 13805702, 286796072, 6727496456, 175903776622, 5073226515772, 160000741383368, 5478160073933490, 202366832844684645, 8022796547785815878, 339769654607776375824, 15309183806159727889536, 731253261602981693567090, 36909816019024064633444820

Offset: 0

Stefano Spezia, Nov 04 2019

nonn

Cf. A007318, A329057, A092392, A000108.

Table[(Binomial[2n,n]Hypergeometric2F1[1,-n,-2n,1+n])/(1+n),{n,0,20}]

{a(n) = sum(k=0,n, binomial(2*n-k, n) * (n+1)^(k-1) )}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Sep 12 2024

{a(n) = my(C = (1 - sqrt(1-4*x +x^2*O(x^n)))/2);
(1/(n+1)) * polcoef( C'/(1 - (n+1)*C), n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Sep 12 2024

a(n) = binomial(2*n, n)*2F1([1, -n], [-2*n], 1 + n)/(1 + n), where 2F1 is the hypergeometric function.
a(n) ~ exp(2) * n^(n-1). - Vaclav Kotesovec, Nov 04 2019
From Paul D. Hanna, Sep 12 2024: (Start)
a(n) = Sum_{k=0..n} binomial(2*n-k, n) * (n+1)^(k-1).
a(n) = (1/(n+1)) * [x^n] C(x)'/(1 - (n+1)*C(x)) for n >= 0 where C(x) = (1 - sqrt(1-4*x))/2 is the g.f. of the Catalan numbers (A000108). (End)

A329058 2-parking triangle T(r, i, 2) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 2 and 0 <= i <= r.

Original entry on oeis.org

1, 2, 1, 7, 6, 3, 30, 36, 32, 16, 143, 220, 275, 250, 125, 728, 1365, 2184, 2808, 2592, 1296, 3876, 8568, 16660, 27440, 36015, 33614, 16807, 21318, 54264, 124032, 248064, 417792, 557056, 524288, 262144, 120175, 346104, 908523, 2133054, 4363065, 7479540, 10097379, 9565938, 4782969

Offset: 0

Stefano Spezia, Nov 02 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  |   2   1
2  |   7   6   3
3  |  30  36  32  16
4  | 143 220 275 250 125

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, A006013, A007318, A329057, A329059, A329060, A329113 (row sums).

T[r_, i_,k_] := (r + 1)^(i-1)*Binomial[k*(r + 1) + r - i - 1, r - i]; Flatten[Table[T[r,i,2],{r,0,9},{i,0,r}]]

T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i).
T(r, 0, 2) = A006013(r).
T(r, r, 2) = A000272(r + 1).

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.

Original entry on oeis.org

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).

A329060 4-parking triangle T(r, i, 4) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 4 and 0 <= i <= r.

Original entry on oeis.org

1, 4, 1, 26, 12, 3, 204, 136, 64, 16, 1771, 1540, 1050, 500, 125, 16380, 17550, 15600, 10800, 5184, 1296, 158224, 201376, 220255, 198940, 139258, 67228, 16807, 1577532, 2324784, 3015936, 3351040, 3063808, 2162688, 1048576, 262144, 16112057, 26978328, 40467492, 53298648, 59960979, 55348596, 39326634, 19131876, 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  |    4      1
2  |   26     12      3
3  |  204    136     64     16
4  | 1771   1540   1050    500    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, A007318, A118971, A329057, A329058, A329059, A329123 (row sums).

T[r_, i_, k_] := (r + 1)^(i-1)*Binomial[k*(r + 1) + r - i - 1, r - i]; Flatten[Table[T[r,i,4],{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, 4) = A118971(r).
T(r, r, 4) = A000272(r + 1).

A331799 Normalized volume of the Caracol flow polytope. Also equal to the number of "unified diagrams" of the Caracol graph (see Section 4.3 and Section 5 in Benedetti et al. reference).

Original entry on oeis.org

1, 3, 32, 625, 18144, 705894, 34603008, 2051893701, 143000000000, 11464341673642, 1039964049506304, 105353940923859082, 11793014101010071552, 1445828316284179687500, 192713711798795989155840, 27750747808814680091687085, 4293818865468117678192721920

Offset: 1

Alejandro H. Morales, Jan 26 2020

			For n=3, a(3) = 32 = 2*(3+1)^2.

C. Benedetti, R. S. González D'León, C. Hanusa, P. E. Harris, A. Khare, A. H. Morales, M. Yip, A combinatorial model for computing volumes of flow polytopes, arXiv:1801.07684 [math.CO], 2018-2019.
C. Benedetti, R. S. González D'León, C. Hanusa, P. E. Harris, A. Khare, A. H. Morales, M. Yip, A combinatorial model for computing volumes of flow polytopes, Trans. Amer. Math. Soc., 372 (2019), 3369-3404.
J. Jang and J. S. Kim, Volumes of flow polytopes related to caracol graphs, arXiv:1911.10703 [math.CO], 2019
M. Yip, A Fuss-Catalan variation of the caracol flow polytope, arXiv:1910.10060 [math.CO], 2019.

Cf. A000108, A000272, A134264.

a:=proc(n)
  return (1/n)*binomial(2*n-2,n-1)*(n+1)^(n-1);
end proc:

Array[(1/#) Binomial[2 # - 2, # - 1] (# + 1)^(# - 1) &, 17] (* Michael De Vlieger, Jan 28 2020 *)

a(n) =  A000108(n-1)*A000272(n+1).
a(n) = (1/n)*binomial(2*n-2,n-1)*(n+1)^(n-1).
a(n) = Sum_{i>=0..n-1} binomial(2*n-2,i)*A329057(n-1,i).

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A329096 Row sums of A329057.

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A329058 2-parking triangle T(r, i, 2) read by rows: T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i) with k = 2 and 0 <= i <= r.

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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.

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A329060 4-parking triangle T(r, i, 4) read by rows: T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i) with k = 4 and 0 <= i <= r.

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A331799 Normalized volume of the Caracol flow polytope. Also equal to the number of "unified diagrams" of the Caracol graph (see Section 4.3 and Section 5 in Benedetti et al. reference).

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A329058 2-parking triangle T(r, i, 2) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 2 and 0 <= i <= r.

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.

A329060 4-parking triangle T(r, i, 4) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 4 and 0 <= i <= r.