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A025015
Central decanomial coefficients: largest coefficient of (1 + x + ... + x^9)^n.
9
1, 1, 10, 75, 670, 6000, 55252, 512365, 4816030, 45433800, 432457640, 4123838279, 39581170420, 380242296850, 3671331273480, 35460394945125, 343900019857310, 3335361909606710, 32458256583753952, 315825118347405835
OFFSET
0,3
COMMENTS
Number of integers in [0, 10^n-1] whose sums of digits are equal to the most common value, which is 9*n/2 for even n and (9*n +/- 1)/2 for odd n > 1. E.g., the most common value of sums of digits of numbers from 0 to 9999 is 9*4/2 = 18, so there are a(4)=670 numbers in this range whose sums of digits are 18. - Warut Roonguthai, Jun 08 2006
Generally, largest coefficient of (1 + x + ... + x^k)^n is asymptotic to (k+1)^n * sqrt(6/(k*(k+2)*Pi*n)). - Vaclav Kotesovec, Aug 09 2013
a(n) is the largest coefficients of the n-th row of A213651. - Miquel Cerda, Jul 19 2017
LINKS
Vaclav Kotesovec, Recurrence
FORMULA
a(n) = Sum_{k=0..floor(9*n/20)}(-1)^(k)*binomial(n, k)*binomial(n+floor(9*n/2)-10*k-1, n-1). - Warut Roonguthai, Jun 08 2006
a(n) ~ 10^n * sqrt(2/(33*Pi*n)). - Vaclav Kotesovec, Aug 09 2013
MATHEMATICA
Flatten[{1, Table[Coefficient[Expand[Sum[x^j, {j, 0, 9}]^n], x^Floor[9*n/2]], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)
KEYWORD
easy,nonn
STATUS
approved