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%I #47 Jul 19 2017 17:08:17
%S 1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,9,9,8,7,6,5,4,3,2,1,1,3,6,10,15,21,
%T 28,36,45,54,61,66,69,70,69,66,61,54,45,36,28,21,15,10,6,3,1,1,4,10,
%U 20,35,56,84,120,165,219,279,342,405,465,519,564,597,615,615,597,564,519,465,405,342,279,219,165,120,84
%N Irregular triangular array T(m,k) with m (row) >= 1 and k (column) >= 1 read by rows: number of m-digit numbers whose digit sum is k.
%C The m-th row is palindromic; T(m,k) = T(m,9*m+1-k).
%H Miquel Cerda, <a href="/A289410/b289410.txt">Rows n=1..10 of triangle, flattened</a>
%F G.f. of row m: (1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m.
%F G.f. as array: (1+x+x^2)*(1+x^3+x^6)*x*y/(1-y*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)). - _Robert Israel_, Jul 19 2017
%e The irregular triangle T(m,k) begins:
%e m\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
%e 1 1 1 1 1 1 1 1 1 1;
%e 2 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1;
%e 3 1 3 6 10 15 21 28 36 45 54 61 66 69 70 69 66 61 54 45,...;
%e 4 1 4 10 20 35 56 84 120 165 219 279 342 405 465,...;
%e 5 1 5 15 35 70 126 210 330 495 714 992 1330 1725,...;
%e 6 1 6 21 56 126 252 462 792 1287 2001 2992,...;
%e etc.
%e Row m(2), column k(4) there are 4 numbers of 2-digits whose digits sum = 4: 13, 22, 31, 40.
%p row:= proc(m) local g; g:= normal((1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m);
%p seq(coeff(g,x,j),j=1..9*m) end proc:
%p seq(row(k),k=1..5); # _Robert Israel_, Jul 19 2017
%Y The row sums = 9*10^(m-1) = A052268(n). The row lengths = 9*m = A008591(n). The middle diagonal = A071976. (row m=3) = A071817, (row m=4) = A090579, (row m=5) = A090580, (row m=6) = A090581, (row m=7) = A278969, (row m=8) = A278971, (row m=9) = A289354, (column k=3) = A000217, (column k=4) = A000292, (column k=5) = A000332, (column k=6) = A000389, (column k=7) = A000579, (column k=8) = A000580, (column k=9) = A000581, (column k=10) = A035927.
%K nonn,base,tabf
%O 1,11
%A _Miquel Cerda_, Jul 05 2017
%E Edited by _Robert Israel_, Jul 19 2017
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