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A165257 Triangle in which n-th row is binomial(n+k-1,k), for column k=1..n.

%I A165257 #20 Dec 21 2024 00:24:55
%S A165257 1,2,3,3,6,10,4,10,20,35,5,15,35,70,126,6,21,56,126,252,462,7,28,84,
%T A165257 210,462,924,1716,8,36,120,330,792,1716,3432,6435,9,45,165,495,1287,
%U A165257 3003,6435,12870,24310
%N A165257 Triangle in which n-th row is binomial(n+k-1,k), for column k=1..n.
%C A165257 T(n,k) is the number of non-descending sequences with length k and last number is less or equal to n. T(n,k) is also the number of integer partitions (of any positive integer) with length k and largest part is less or equal to n. - _Zlatko Damijanic_, Dec 06 2024
%H A165257 Harvey P. Dale, <a href="/A165257/b165257.txt">Table of n, a(n) for n = 1..1000</a>
%e A165257 1;
%e A165257 2, 3;
%e A165257 3, 6, 10;
%e A165257 4, 10, 20, 35;
%e A165257 5, 15, 35, 70, 126;
%e A165257 6, 21, 56, 126, 252, 462;
%e A165257 7, 28, 84, 210, 462, 924, 1716;
%t A165257 Table[Binomial[n+k-1,k],{n,10},{k,n}]//Flatten (* _Harvey P. Dale_, Jul 31 2021 *)
%o A165257 (PARI) tabl(nn) = {for (n=1, nn, for (k=1, n, print1( binomial(n+k-1, k), ", ");); print(););} \\ _Michel Marcus_, Jun 12 2013
%Y A165257 A059481 with the first column (k = 0) removed.
%Y A165257 Cf. A030662 (row sums), A001700 (diagonal).
%K A165257 easy,nonn,tabl
%O A165257 1,2
%A A165257 Daniel McLaury (daniel.mclaury(AT)gmail.com), Sep 11 2009