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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.

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 * A189503 A190076 A278729

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 August 18 22:16 EDT 2017. Contains 290768 sequences.