A329058 -id:A329058 - cp's OEIS frontend

A329113 Row sums of A329058.

1, 3, 16, 114, 1013, 10973, 142980, 2208958, 39796747, 822252765, 19192927840, 499725372072, 14361189358332, 451551559895244, 15419952421504604, 568327808746891014, 22486286328779957223, 950624702855155671101, 42765431507866487733504, 2039838152324158031743570

Offset: 0

Stefano Spezia, Nov 05 2019

nonn

Cf. A007318, A329058.

Table[(Binomial[1+3n,n]Hypergeometric2F1[1,-n,-1-3n,1+n])/(1+n),{n,0,19}]

a(n) = binomial(1 + 3*n, n)*2F1([1, -n], [-1 - 3*n], 1 + n)/(1 + n), where 2F1 is the hypergeometric function.

a(n) ~ exp(3) * n^(n-1). - Vaclav Kotesovec, Nov 06 2019

A329057 1-parking triangle T(r, i, 1) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 1 and 0 <= i <= r.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 10, 16, 16, 14, 35, 75, 125, 125, 42, 126, 336, 756, 1296, 1296, 132, 462, 1470, 4116, 9604, 16807, 16807, 429, 1716, 6336, 21120, 61440, 147456, 262144, 262144, 1430, 6435, 27027, 104247, 360855, 1082565, 2657205, 4782969, 4782969, 4862, 24310, 114400, 500500, 2002000, 7150000, 22000000, 55000000, 100000000, 100000000

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  |  1   1
2  |  2   3   3
3  |  5  10  16  16
4  | 14  35  75 125 125

Stefano Spezia, First 151 rows of the triangle, flattened
Carolina Benedetti, Rafael S. González D’León, Christopher R. H. Hanusa, Pamela E. Harris, Apoorva Khare, Alejandro H. Morales, Martha Yip, The volume of the caracol polytope, Séminaire Lotharingien de Combinatoire 80B.87 (2018).
Martha Yip, A Fuss-Catalan variation of the caracol flow polytope, arXiv:1910.10060 [math.CO], 2019.

Cf. A000108, A000272, A007318, A329058, A329059, A329060, A329096 (row sums).

T[r_, i_,k_] := (r + 1)^(i-1)*Binomial[k*(r + 1) + r - i - 1, r - i]; Flatten[Table[T[r,i,1],{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, 1) = A000108(r).
T(r, r, 1) = 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).

cp's OEIS Frontend

A329113 Row sums of A329058.

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A329057 1-parking triangle T(r, i, 1) read by rows: T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i) with k = 1 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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Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A329057 1-parking triangle T(r, i, 1) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 1 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.