A123297 -id:A123297 - cp's OEIS frontend

A372307 Square array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 9, 10, 1, 0, 1, 1, 44, 297, 56, 1, 0, 1, 1, 265, 13756, 13833, 346, 1, 0, 1, 1, 1854, 925705, 6699824, 748521, 2252, 1, 0, 1, 1, 14833, 85394646, 5691917785, 3993445276, 44127009, 15184, 1, 0, 1

Offset: 0

Jeremy Tan, Apr 26 2024

A deck has k suits of n cards each. The deck is shuffled and dealt into k hands of n cards each. A match occurs for every card in the i-th hand of suit i. T(n,k) is the number of ways of achieving no matches. The probability of no matches is T(n,k)/((n*k)!/n!^k).

T(n,k) is the maximal number of totally mixed Nash equilibria in games of k players, each with n+1 pure options.

			Square array T(n,k) begins:
  1, 1, 1,      1,            1,                   1, ...
  1, 0, 1,      2,            9,                  44, ...
  1, 0, 1,     10,          297,               13756, ...
  1, 0, 1,     56,        13833,             6699824, ...
  1, 0, 1,    346,       748521,          3993445276, ...
  1, 0, 1,   2252,     44127009,       2671644472544, ...
  1, 0, 1,  15184,   2750141241,    1926172117389136, ...
  1, 0, 1, 104960, 178218782793, 1463447061709156064, ...

Jeremy Tan, Antidiagonals n = 0..32, flattened
Shalosh B. Ekhad, Christoph Koutschan and Doron Zeilberger, There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts], arXiv:2101.10147 [math.CO], 2021.
S. Even and J. Gillis, Derangements and Laguerre polynomials, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 79, Issue 1, January 1976, pp. 135-143.
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), pp. 107-118.
Jeremy Tan, Python program
Raimundas Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arXiv:1401.5400 [math.CO], 2014.
Raimundas Vidunas, Counting derangements and Nash equilibria, Ann. Comb. 21, No. 1, 131-152 (2017).
Index entries for sequences related to card matching

Rows 0-5 give A000012, A000166, A000459, A059073, A059074, A123297.
Columns 0-4 give A000012, A000007, A000012, A000172, A371252.
Main diagonal gives A375778.

A:= (n, k)-> (-1)^(n*k)*int(exp(-x)*orthopoly[L](n, x)^k, x=0..infinity):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Aug 27 2024

Table[Abs[Integrate[Exp[-x] LaguerreL[n, x]^(s-n), {x, 0, Infinity}]], {s, 0, 9}, {n, 0, s}] // Flatten

Python
```
# See link.
```

T(n,k) = (-1)^(n*k) * Integral_{x=0..oo} exp(-x)*L_n(x)^k dx, where L_n(x) is the Laguerre polynomial of degree n (Even and Gillis).

T(n,k) ~ A089759(n,k)/exp(n).

A123294 Sum of 13 positive 5th powers.

Original entry on oeis.org

13, 44, 75, 106, 137, 168, 199, 230, 255, 261, 286, 292, 317, 323, 348, 354, 379, 385, 410, 416, 441, 472, 497, 503, 528, 534, 559, 565, 590, 596, 621, 627, 652, 683, 714, 739, 745, 770, 776, 801, 807, 832, 838, 863, 894, 925, 956, 981, 987, 1012, 1018, 1036

Offset: 1

Jonathan Vos Post, Sep 24 2006

Up to 416 = 13*(2^5) this sequence is identical to x+1 for x in A003357 Sum of 12 positive 5th powers. Primes in this sequence (13, 137, 199, 317, ...) are A123299. As proved by J.-R. Chen in 1964, g(5) = 37, so every positive integer can be written as the sum of no more than 37 positive 5th powers. G(5) <= 17, bounding the least integer G(5) such that every positive integer beyond a certain point (i.e., all but a finite number) is the sum of G(5) 5th powers.

			a(1) = 13 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 44 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5.
a(9) = 255 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
a(11) = 286 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123295-A123297, A008480, A123299.

up = 1500; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 13}]; a (* Giovanni Resta, Jun 12 2016 *)

Sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.

Two missing terms and more terms from Giovanni Resta, Jun 12 2016

A123295 Sum of 14 positive 5th powers.

Original entry on oeis.org

14, 45, 76, 107, 138, 169, 200, 231, 256, 262, 287, 293, 318, 324, 349, 355, 380, 386, 411, 417, 442, 448, 473, 498, 504, 529, 535, 560, 566, 591, 597, 622, 628, 653, 659, 684, 715, 740, 746, 771, 777, 802, 808, 833, 839, 864, 870, 895, 926, 957, 982, 988

Offset: 1

Jonathan Vos Post, Sep 24 2006

Up to 417 = 13*(2^5) + 1 this sequence is identical to x+2 for x in A003357 Sum of 12 positive 5th powers. Primes in this sequence (107, 293, 349, 653, ...) are A123300. As proved by J.-R. Chen in 1964, g(5) = 37, so every positive integer can be written as the sum of no more than 37 positive 5th powers. G(5) <= 17, bounding the least integer G(5) such that every positive integer beyond a certain point (i.e., all but a finite number) is the sum of G(5) 5th powers.

			a(1) = 14 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 45 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5.
a(9) = 256 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
a(11) = 287 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123294-A123297, A008480, A123299, A123300.

up = 1000; q = Range[up^(1/5)]^5; a ={0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a=b, {k, 14}]; a (* Giovanni Resta, Jun 12 2016 *)

Sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.

5 missing terms and more terms from Giovanni Resta, Jun 12 2016

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A372307 Square array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.

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A123294 Sum of 13 positive 5th powers.

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A123295 Sum of 14 positive 5th powers.

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Mathematica

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Extensions