A265080 -id:A265080 - cp's OEIS frontend

A265081 Row 4 of array in A265080.

0, 4, 44, 192, 544, 1220, 2364, 4144, 6752, 10404, 15340, 21824, 30144, 40612, 53564, 69360, 88384, 111044, 137772, 169024, 205280, 247044, 294844, 349232, 410784, 480100, 557804, 644544, 740992, 847844, 965820, 1095664, 1238144, 1394052, 1564204, 1749440, 1950624

Offset: 0

N. J. A. Sloane, Jan 01 2016

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
D. R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.

Cf. A265080.

a(6) onwards from Andrew Howroyd, Aug 09 2025

A265082 Row 5 of array in A265080.

Original entry on oeis.org

0, 5, 110, 675, 2540, 7145, 16650, 34055, 63320, 109485, 178790, 278795, 418500, 608465, 860930, 1189935, 1611440, 2143445, 2806110, 3621875, 4615580, 5814585, 7248890, 8951255, 10957320, 13305725, 16038230, 19199835, 22838900, 27007265, 31760370

Offset: 0

N. J. A. Sloane, Jan 01 2016

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
D. R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.

Cf. A265080.

a(6) onwards from Andrew Howroyd, Aug 09 2025

A265083 Column 3 of array in A265080.

Original entry on oeis.org

0, 3, 12, 51, 192, 675, 2358, 8043, 26736, 88659, 290910, 941985, 3046068, 9791613, 31221414, 99466515, 315677664, 996398883, 3143147598, 9888728505, 30988353300, 97066250379, 303448558578, 945804661377, 2946961001304, 9168357598425, 28456994858058, 88302639334743

Offset: 0

N. J. A. Sloane, Jan 01 2016

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500
D. R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.

Cf. A265080.

seq(n)={Vec(serlaplace(sum(j=0, n, exp(3*x + O(x*x^n)) - sum(i=0, j, x^i/i!, O(x*x^n))^3)), -n-1)} \\ Andrew Howroyd, Aug 09 2025

E.g.f.: Sum_{j>=0} (exp(x)^3 - (Sum_{i=0..j} x^i/i!)^3). - Andrew Howroyd, Aug 09 2025

a(6) onwards from Andrew Howroyd, Aug 09 2025

A265084 Column 4 of array in A265080.

Original entry on oeis.org

0, 4, 20, 108, 544, 2540, 11544, 52192, 231872, 1014444, 4401400, 19016888, 81588720, 347833408, 1476460496, 6250982640, 26366866432, 110825294252, 464575219704, 1944165731800, 8117172816560, 33813868882584, 140605002878032, 583960731640688, 2421457649730528, 10025040350809600

Offset: 0

N. J. A. Sloane, Jan 01 2016

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500
D. R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.

Cf. A265080.

seq(n)={Vec(serlaplace(sum(j=0, n, exp(4*x + O(x*x^n)) - sum(i=0, j, x^i/i!, O(x*x^n))^4)), -n-1)} \\ Andrew Howroyd, Aug 09 2025

E.g.f.: Sum_{j>=0} (exp(x)^4 - (Sum_{i=0..j} x^i/i!)^4). - Andrew Howroyd, Aug 09 2025

a(6) onwards from Andrew Howroyd, Aug 09 2025

A265085 Column 5 of array in A265080.

Original entry on oeis.org

0, 5, 30, 195, 1220, 7145, 40230, 224175, 1241000, 6785325, 36675650, 196823495, 1051855260, 5599171045, 29659801150, 156443274375, 822605274320, 4315160956565, 22584570699690, 117904543416615, 614153114242100, 3193305076085145, 16578603268278590, 85946023037719835

Offset: 0

N. J. A. Sloane, Jan 01 2016

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500
D. R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.

Cf. A265080.

a(6) onwards from Andrew Howroyd, Aug 09 2025

A208250 The sum of the largest preimage over all functions f:{1,2,...,n}->{1,2,...,n}.

Original entry on oeis.org

0, 1, 6, 51, 544, 7145, 112356, 2066323, 43574336, 1036922769, 27486891100, 803137535321, 25642631336400, 888148407804853, 33165208812574216, 1328185604750416875, 56783630865774075136, 2581268127178259819297, 124322489582200453748268, 6324172127062894070727625

Offset: 0

Geoffrey Critzer, Jan 15 2013

nonn

n labeled balls are placed in n labeled urns. The maximum number of balls in an urn is summed over all n^n possible configurations. a(n) is this sum.

			a(2) = 6.  The functions f:{1,2}->{1,2} written as words are: 11, 12, 21, 22 and we sum respectively 2 + 1 + 1 + 2 = 6.

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison and Wesley, 1996, page 435.

Robert Gerbicz, Table of n, a(n) for n = 0..386
D. R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.
Sela Fried, The expected degree of noninvertibility of compositions of functions and a related combinatorial identity, arXiv:2202.13061 [math.CO], 2022.
Robert Gerbicz, a(n) for n = 0..1024 (an a-file)
Robert Gerbicz, gmp code
G. H. Gönnet, Expected length of the longest probe sequence in hash code searching, Journal of the ACM, 28:2 (1981), pp. 289-304.
Michael Mitzenmacher, Andréa W. Richa, and Ramesh Sitaraman, The Power of Two Random Choices: A Survey of Techniques and Results

Main diagonal of A265080.

f[n_] := n! Coefficient[ Series[ Sum[ Exp[n*x] - Sum[x^i/i!, {i, 0, j}]^n, {j, 0, n}], {x, 0, n}], x^n]; f[0] = 0; Array[f, 19, 0] (* modified by Robert G. Wilson v, Feb 20 2013 *)

a(n) = n! * [x^n] Sum_{j>=0} (exp(x)^n - (Sum_{i=0..j} x^i/i!)^n).

a(n) ~ n^n log n/log log n. More precisely, a(n)/n^n = Gamma^(-1)(n) - 3/2 + o(1) where Gamma^(-1) is the inverse of the gamma function. See Gönnet section 4 or Mitzenmacher et al. - Charles R Greathouse IV, Feb 20 2013

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A265081 Row 4 of array in A265080.

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A265082 Row 5 of array in A265080.

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A265083 Column 3 of array in A265080.

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A265084 Column 4 of array in A265080.

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A265085 Column 5 of array in A265080.

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A208250 The sum of the largest preimage over all functions f:{1,2,...,n}->{1,2,...,n}.

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