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%I #27 Feb 29 2020 15:14:54
%S 1,2,6,20,70,252,924,3432,12870,48619,184735,705222,2702609,10390940,
%T 40062132,154830696,599641425,2326640877,9042327525,35194002709,
%U 137160956815,535193552973,2090558951396,8174176541450,31990402045260
%N Number of lists of length n from {0..9} summing to n but not beginning with 0.
%C Number of n-digit numbers with digit sum n.
%C Middle diagonal of A213651. - _Miquel Cerda_, Aug 11 2017
%H Robert Israel, <a href="/A071976/b071976.txt">Table of n, a(n) for n = 1..1100</a>
%H Ji Young Choi, <a href="https://www.emis.de/journals/JIS/VOL22/Choi/choi15.html">Digit Sums Generalizing Binomial Coefficients</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.3.
%F 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.]
%F Coefficient of x^n in T^n - T^(n-1), where T = 1+x+...+x^9. - _Robert Israel_, Apr 06 2016
%e a(3) = 6 as there are six three-digit numbers with digit sum 3: 102, 111, 120, 201, 210, 300.
%e a(10) = binomial(18,9)-1; a(11) = binomial(20,10)-21; a(12) = binomial(22,11)-210.
%p T:= add(x^i,i=0..9):
%p seq(coeff(T^n - T^(n-1),x,n), n=1..25); # _Robert Israel_, Apr 06 2016
%t 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}]
%o (PARI) a(n)=local(y=(x^10-1)/(x-1)); if(n<1,0,polcoeff((y-1)*y^(n-1),n))
%Y Different from A000984.
%Y Number of n-digit entries in A061384.
%K nonn,base
%O 1,2
%A _Amarnath Murthy_, Jun 18 2002
%E Edited by _N. J. A. Sloane_ and _Robert G. Wilson v_, Jun 20 2002
%E More terms from _Vladeta Jovovic_, Jun 21 2002
%E More terms from _John W. Layman_, Jun 22 2002
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