A002879 -id:A002879 - cp's OEIS frontend

A328647 Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (1+x)/(x^2-3x+1)).

1, 1, 4, -2, -1, 11, -12, 3, 1, 29, -44, 24, -4, -1, 76, -145, 110, -40, 5, 1, 199, -456, 435, -220, 60, -6, -1, 521, -1393, 1596, -1015, 385, -84, 7, 1, 1364, -4168, 5572, -4256, 2030, -616, 112, -8, -1, 3571, -12276, 18756, -16716, 9576, -3654, 924, -144

Offset: 0

Clark Kimberling, Nov 01 2019

The first 201 polynomials are irreducible. Column 1 of the array: A002879 (odd-indexed Lucas numbers). Row sums: A000032 (Lucas numbers). Alternating row sums: essentially 5*A030191.

			First eight rows:
     1,     1;
     4,    -2,   -1;
    11,   -12,    3,     1;
    29,   -44,   24,    -4,   -1;
    76,  -145,  110,   -40,    5,    1;
   199,  -456,  435,  -220,   60,   -6,  -1;
   521, -1393, 1596, -1015,  385,  -84,   7,  1;
  1364, -4168, 5572, -4256, 2030, -616, 112, -8, -1;
First eight polynomials:
1 + x
4 - 2 x - x^2
11 - 12 x + 3 x^2 + x^3
29 - 44 x + 24 x^2 - 4 x^3 - x^4
76 - 145 x + 110 x^2 - 40 x^3 + 5 x^4 + x^5
199 - 456 x + 435 x^2 - 220 x^3 + 60 x^4 - 6 x^5 - x^6
521 - 1393 x + 1596 x^2 - 1015 x^3 + 385 x^4 - 84 x^5 + 7 x^6 + x^7
1364 - 4168 x + 5572 x^2 - 4256 x^3 + 2030 x^4 - 616 x^5 + 112 x^6 - 8 x^7 - x^8

Cf. A328646, A002879, A000032.

g[x_, n_] := Numerator[ Factor[D[(1 + x)/(x^2 - 3 x + 1), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x]
Table[h[n], {n, 0, 10}]
Column[%]  (* A328647 array *)

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

Original entry on oeis.org

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

Offset: 1

Gerard P. Michon, Aug 06 2008

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.

			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.

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.

Cf. A002761, A002806, A002834, A002879.

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.

cp's OEIS Frontend

A328647 Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (1+x)/(x^2-3x+1)).

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