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A264751 Triangle read by rows: T(n,k) is the number of sequences of k <= n throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the k-th throw.

%I A264751 #75 Mar 26 2021 19:26:40
%S A264751 1,1,2,1,5,3,1,9,11,4,1,14,26,19,5,1,20,50,55,29,6,1,27,85,125,99,41,
%T A264751 7,1,35,133,245,259,161,55,8,1,44,196,434,574,476,244,71,9,1,54,276,
%U A264751 714,1134,1176,804,351,89,10,1,65,375,1110,2058,2562,2190,1275,485,109,11
%N A264751 Triangle read by rows: T(n,k) is the number of sequences of k <= n throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the k-th throw.
%C A264751 By empirical observation: Sum of rows is A002064.
%H A264751 Cyann Donnot, Antoine Genitrini, Yassine Herida, <a href="https://hal.sorbonne-universite.fr/hal-02462764">Unranking Combinations Lexicographically: an efficient new strategy compared with others</a>, hal-02462764 [cs] / [cs.DS] / [math] / [math.CO], 2020.
%H A264751 Antoine Genitrini and Martin Pépin, <a href="https://hal.sorbonne-universite.fr/hal-03040740v2">Lexicographic unranking of combinations revisited</a>, hal-03040740v2 [cs.DM] [cs.DS] [math.CO], 2020.
%F A264751 Sum_{k = 1..n} T(n,k)*k/n^k = ((n+1)/n)^(n-1) = expected value of k.
%F A264751 Lim_{n->infinity} (expected value of k) = e = 2.71828182845... - _Jon E. Schoenfield_, Nov 26 2015
%F A264751 T(n,k) = Sum_{i=k..n} i*binomial(i-2,k-2). - _Danny Rorabaugh_, Mar 04 2016
%F A264751 T(n,n-1) = 2*T(n-1,n-1) + T(n-1,n-2).
%F A264751 By empirical observation, g.f. for column k: (x-k)/(x-1)^(k+1).
%e A264751 Triangle begins:
%e A264751   1
%e A264751   1    2
%e A264751   1    5    3
%e A264751   1    9   11    4
%e A264751   1   14   26   19    5
%e A264751   1   20   50   55   29    6
%e A264751   1   27   85  125   99   41    7
%e A264751   1   35  133  245  259  161   55    8
%e A264751   1   44  196  434  574  476  244   71    9
%e A264751   1   54  276  714 1134 1176  804  351   89   10
%e A264751   1   65  375 1110 2058 2562 2190 1275  485  109   11
%t A264751 T[n_, k_] := Module[
%t A264751 {i, total = 0, part, perm},
%t A264751 part = IntegerPartitions[n, {k}];
%t A264751 perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}],      1];
%t A264751 For[i = 1, i <= Length[perm], i++,    total += n + 1 - perm[[i, k]]    ];
%t A264751 Return[total];   ]
%t A264751 (* The rows are obtained by: *)
%t A264751 g[n_] := Table[T[n,k], {k,1,n}]
%t A264751 (* And the triangle is obtained by: *)
%t A264751 Table[g[n],{n,1,number_of_rows_wanted}]
%Y A264751 Columns are: A000012 (k=1), A000096 (k=2), A051925 (k=3), A215862 (k=4), A264750 (k=5).
%Y A264751 Cf. A007318 (binomial(n-1,k-1) = number of sequences of k throws of an n-sided die in which the sum of the throws equals n).
%Y A264751 See also A002064.
%K A264751 nonn,tabl
%O A264751 1,3
%A A264751 _Louis Rogliano_, Nov 26 2015