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A289410
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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.
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3
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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
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OFFSET
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1,11
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COMMENTS
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The m-th row is palindromic; T(m,k) = T(m,9*m+1-k).
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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:
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CROSSREFS
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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.
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KEYWORD
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nonn,base,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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