A289410 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.
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, 28, 36, 45, 54, 61, 66, 69, 70, 69, 66, 61, 54, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 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
Offset: 1
Examples
The irregular triangle T(m,k) begins: m\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 1 1 1 1 1 1 1 1 1; 2 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1; 3 1 3 6 10 15 21 28 36 45 54 61 66 69 70 69 66 61 54 45,...; 4 1 4 10 20 35 56 84 120 165 219 279 342 405 465,...; 5 1 5 15 35 70 126 210 330 495 714 992 1330 1725,...; 6 1 6 21 56 126 252 462 792 1287 2001 2992,...; etc. Row m(2), column k(4) there are 4 numbers of 2-digits whose digits sum = 4: 13, 22, 31, 40.
Links
- Miquel Cerda, Rows n=1..10 of triangle, flattened
Crossrefs
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.
Programs
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Maple
row:= proc(m) local g; g:= normal((1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m); seq(coeff(g,x,j),j=1..9*m) end proc: seq(row(k),k=1..5); # Robert Israel, Jul 19 2017
Formula
G.f. of row m: (1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m.
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
Extensions
Edited by Robert Israel, Jul 19 2017
Comments