A143314 Number of hands of n cards containing a straight flush (for n=1 to 52).

0, 0, 0, 0, 40, 1844, 41584, 611340, 6588116, 55482100, 380126920, 2177910310, 10644616240, 45049914588, 167011924492, 547315800984, 1597026077496, 4173458163098, 9813490226056, 20841357619302, 40096048882028

Offset: 1

Author

Gerard P. Michon, Aug 06 2008

Keywords

• fini
• full
• nonn

Comments

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).

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

Examples

			a(5) = 40 because each suit allows 10 straight flushes (2 of which contain an ace).
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).
a(45) = 133784560 = C(52,45) because every hand of 45 cards (or more) includes a straight flush.
a(52) = 1 because there's only one "hand" of 52 cards.
		

Links

• Gerard P. Michon, Aug 06 2008, Table of n, a(n) for n = 1..52
• G. P. Michon, q-Card Poker.
• Brian Wu and Chai Wah Wu, Big Two and n-card poker probabilities, arXiv:2309.00011 [math.HO], 2023.

Crossrefs

Cf. A002761, A002806, A002834, A002879.

Formula

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.