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

Offset

1,2

Comments

Number of n-digit numbers with digit sum n.

Middle diagonal of A213651. - Miquel Cerda, Aug 11 2017

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.

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

Example

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.

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

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}]

Prog

(PARI) a(n)=local(y=(x^10-1)/(x-1)); if(n<1, 0, polcoeff((y-1)*y^(n-1), n))

Crossrefs

Different from A000984.

Number of n-digit entries in A061384.

Sequence in context: A056616 A065346 A302645 * A302646 A000984 A087433

Adjacent sequences: A071973 A071974 A071975 * A071977 A071978 A071979

Keyword

nonn,base

Author

Amarnath Murthy, Jun 18 2002

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

Status

approved