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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
Sequence in context: A145294 A147520 A190926 * A308836 A189503 A190076
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.)