%I #13 Sep 04 2023 01:44:12

%S 0,0,0,0,40,1844,41584,611340,6588116,55482100,380126920,2177910310,

%T 10644616240,45049914588,167011924492,547315800984,1597026077496,

%U 4173458163098,9813490226056,20841357619302,40096048882028

%N Number of hands of n cards containing a straight flush (for n=1 to 52).

%C With a regular deck of 52 playing cards (4 suits of 13 cards: 23456789TJQKA) a "straight flush" consists of 5 cards of the same suit with consecutive values. The ace (A) is considered to come either before the deuce (2) or after the king (K).

%C The first terms of the sequence are zero because there are no straight flushes in a hand of fewer than 5 cards.

%H Gerard P. Michon, Aug 06 2008, <a href="/A143314/b143314.txt">Table of n, a(n) for n = 1..52</a>

%H G. P. Michon, <a href="http://www.numericana.com/answer/counting.htm#stud26">q-Card Poker</a>.

%H Brian Wu and Chai Wah Wu, <a href="https://arxiv.org/abs/2309.00011">Big Two and n-card poker probabilities</a>, arXiv:2309.00011 [math.HO], 2023.

%F The generating function is a polynomial: (1+x)^52 - ((1+x)^13 - x^5(1+x)(10 + 61x + 156x^2 + 215x^3 + 169x^4 + 65x^5 + 12x^6 + x^7))^4.

%e a(5) = 40 because each suit allows 10 straight flushes (2 of which contain an ace).

%e a(44) = 752538149 = C(52,44) - 1 because there's only one way to avoid a straight flush with 44 cards (namely, 2346789JQKA in every suit).

%e a(45) = 133784560 = C(52,45) because every hand of 45 cards (or more) includes a straight flush.

%e a(52) = 1 because there's only one "hand" of 52 cards.

%Y Cf. A002761, A002806, A002834, A002879.

%K fini,full,nonn

%O 1,5

%A _Gerard P. Michon_, Aug 06 2008