A071976 Number of lists of length n from {0..9} summing to n but not beginning with 0.
1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48619, 184735, 705222, 2702609, 10390940, 40062132, 154830696, 599641425, 2326640877, 9042327525, 35194002709, 137160956815, 535193552973, 2090558951396, 8174176541450, 31990402045260, 125301956523471, 491168514123342
Offset: 1
Examples
a(3) = 6 as there are six three-digit numbers with digit sum 3: 102, 111, 120, 201, 210, 300. a(10) = binomial(18,9)-1; a(11) = binomial(20,10)-21; a(12) = binomial(22,11)-210.
Links
- Robert Israel, Table of n, a(n) for n = 1..1100
- Ji Young Choi, Digit Sums Generalizing Binomial Coefficients, J. Int. Seq., Vol. 22 (2019), Article 19.8.3.
Programs
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Maple
T:= add(x^i,i=0..9): seq(coeff(T^n - T^(n-1),x,n), n=1..25); # Robert Israel, Apr 06 2016
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Mathematica
Do[c = 0; k = 10^n; l = 10^(n + 1) - 1; While[k < l, If[ Plus @@ IntegerDigits[k] == n + 1, c++ ]; k++ ]; Print[c], {n, 0, 7}] -
PARI
a(n)=local(y=(x^10-1)/(x-1)); if(n<1,0,polcoeff((y-1)*y^(n-1),n))
Formula
Equals binomial(2n-2, n-1) for n <= 9, by the stars and bars argument. [To get such a number take n boxes in which the leftmost box contains a 1 and the rest are empty. Distribute the remaining n-1 1's into the n boxes subject to the constraint that no box contains more than 9 1's. This can be done in binomial(2n-2, n-1) ways for n <= 9.]
Coefficient of x^n in T^n - T^(n-1), where T = 1+x+...+x^9. - Robert Israel, Apr 06 2016
Extensions
Edited by N. J. A. Sloane and Robert G. Wilson v, Jun 20 2002
More terms from Vladeta Jovovic, Jun 21 2002
More terms from John W. Layman, Jun 22 2002
Comments