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Search: id:A143314
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| A143314 |
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Number of hands of n cards containing a straight flush (for n=1 to 52). |
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+0
1
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| 0, 0, 0, 0, 40, 1844, 41584, 611340, 6588116, 55482100, 380126920, 2177910310, 10644616240, 45049914588, 167011924492, 547315800984, 1597026077496, 4173458163098, 9813490226056, 20841357619302, 40096048882028
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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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.
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LINKS
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Gerard P. Michon (g.michon(AT)att.net), Aug 06 2008, Table of n, a(n) for n = 1..52
G. P. Michon, q-Card Poker.
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FORMULA
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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
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EXAMPLE
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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 decause there's only one "hand" of 52 cards.
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CROSSREFS
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Cf. A002761, A002806, A002834, A002879.
Sequence in context: A140702 A145294 A147520 this_sequence A060056 A140729 A003741
Adjacent sequences: A143311 A143312 A143313 this_sequence A143315 A143316 A143317
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KEYWORD
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fini,full,nonn
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AUTHOR
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Gerard P. Michon (g.michon(AT)att.net), Aug 06 2008
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