

A264750


Number of sequences of 5 throws of an nsided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the 5th throw.


2



5, 29, 99, 259, 574, 1134, 2058, 3498, 5643, 8723, 13013, 18837, 26572, 36652, 49572, 65892, 86241, 111321, 141911, 178871, 223146, 275770, 337870, 410670, 495495, 593775, 707049, 836969, 985304, 1153944, 1344904, 1560328, 1802493, 2073813, 2376843, 2714283
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OFFSET

5,1


COMMENTS

Sequence gives the second column of A185508. [Bruno Berselli, Nov 24 2015]
Number of 5tuples (t_1, ..., t_5) with 1 <= t_j <= n, Sum_{j <= 4} t_j < n and Sum_{j<=5} t_j >= n.  Robert Israel, Nov 25 2015


LINKS

Colin Barker, Table of n, a(n) for n = 5..1000
Louis Rogliano, Sequence A264750
Index entries for linear recurrences with constant coefficients, signature (6,15,20,15,6,1).


FORMULA

From Colin Barker, Nov 23 2015: (Start)
a(n) = (n  4)*(n  3)*(n  2)*(n  1)*(4*n + 5)/120.
a(n) = 6*a(n1)  15*a(n2) + 20*a(n3)  15*a(n4) + 6*a(n5)  a(n6) for n>10.
G.f.: x^5*(5  x) / (1  x)^6. (End)


EXAMPLE

From Jon E. Schoenfield, Nov 26 2015: (Start)
For n=5, the a(5) = 5 sequences (i.e., quintuples or 5tuples) are {1,1,1,1,1}, {1,1,1,1,2}, {1,1,1,1,3}, {1,1,1,1,4} and {1,1,1,1,5}. (Each of the first four throws must be a 1; otherwise, the sum of the throws would reach or exceed 5 before the 5th throw.)
For n=6, each of the quintuples must have four throws whose sum is less than 6, followed by a fifth throw that brings the sum to at least 6, so the a(6) = 29 quintuples are the 5 quintuples {1,1,1,1,t_5} where t_5 is any value in 2..6 and the four sets of 6 quintuples {1,1,1,2,t_5}, {1,1,2,1,t_5}, {1,2,1,1,t_5} and {2,1,1,1,t_5} where t_5 is any value in 1..6. (End)


MAPLE

A264750:=n>(n4)*(n3)*(n2)*(n1)*(4*n+5)/120: seq(A264750(n), n=5..50); # Wesley Ivan Hurt, Nov 24 2015


MATHEMATICA

f[n_, k_] := Module[
{i, total = 0, part, perm},
part = IntegerPartitions[n, {k}];
perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}], 1];
For[i = 1, i <= Length[perm], i++, total += n + 1  perm[[i, k]] ];
Return[total]; ]
And the sequences are obtained by:
h[k_] := Table[f[i, k], {i, k, number_of_terms_wanted}]
Table[(n  4) (n  3) (n  2) (n  1) (4 n + 5)/120, {n, 5, 40}] (* Bruno Berselli, Nov 24 2015 *)


PROG

(PARI) Vec(x^5*(5x)/(1x)^6 + O(x^100)) \\ Colin Barker, Nov 23 2015
(PARI) for(n=5, 40, print1((n4)*(n3)*(n2)*(n1)*(4*n+5)/120", ")); \\ Bruno Berselli, Nov 24 2015
(MAGMA) [(n4)*(n3)*(n2)*(n1)*(4*n+5)/120: n in [5..40]]; // Vincenzo Librandi, Nov 24 2015
(Sage) [(n4)*(n3)*(n2)*(n1)*(4*n+5)/120 for n in (5..40)] # Bruno Berselli, Nov 24 2015


CROSSREFS

Cf. A000096 (k=2), A051925 (k=3), A215862 (k=4).
Cf. A185508.
Sequence in context: A190585 A197276 A211062 * A205172 A139856 A097345
Adjacent sequences: A264747 A264748 A264749 * A264751 A264752 A264753


KEYWORD

nonn,easy


AUTHOR

Louis Rogliano, Nov 23 2015


EXTENSIONS

Offset changed by Robert Israel, Nov 25 2015
Formulae, bfile adapted to the new offset and definition rephrased by the Editors of the OEIS, Nov 26 2015


STATUS

approved



