OFFSET
1,11
COMMENTS
The m-th row is palindromic; T(m,k) = T(m,9*m+1-k).
LINKS
Miquel Cerda, Rows n=1..10 of triangle, flattened
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
EXAMPLE
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.
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
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.
KEYWORD
nonn,base,tabf
AUTHOR
Miquel Cerda, Jul 05 2017
EXTENSIONS
Edited by Robert Israel, Jul 19 2017
STATUS
approved