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 A143314 Number of hands of n cards containing a straight flush (for n=1 to 52). 1
 0, 0, 0, 0, 40, 1844, 41584, 611340, 6588116, 55482100, 380126920, 2177910310, 10644616240, 45049914588, 167011924492, 547315800984, 1597026077496, 4173458163098, 9813490226056, 20841357619302, 40096048882028 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 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. 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. 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. EXAMPLE 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. CROSSREFS Cf. A002761, A002806, A002834, A002879. Sequence in context: A145294 A147520 A190926 * A308836 A189503 A190076 Adjacent sequences: A143311 A143312 A143313 * A143315 A143316 A143317 KEYWORD fini,full,nonn AUTHOR Gerard P. Michon, Aug 06 2008 STATUS approved

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Last modified May 23 21:14 EDT 2024. Contains 372765 sequences. (Running on oeis4.)