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A265080 Array read by antidiagonals, arising from study of remixing keys in public-key cryptography.

%I A265080 #32 Aug 10 2025 02:50:09
%S A265080 0,0,0,0,1,0,0,2,2,0,0,3,6,3,0,0,4,12,18,4,0,0,5,20,51,44,5,0,0,6,30,
%T A265080 108,192,110,6,0,0,7,42,195,544,675,252,7,0,0,8,56,318,1220,2540,2358,
%U A265080 588,8,0,0,9,72,483,2364,7145,11544,8043,1304,9,0
%N A265080 Array read by antidiagonals, arising from study of remixing keys in public-key cryptography.
%C A265080 See Brown (2015) for precise definition.
%C A265080 If you randomly throw n balls into k boxes then T(n,k)/k^n is the expected number of balls in the fullest box. - _Henry Bottomley_, Mar 20 2021
%H A265080 Andrew Howroyd, <a href="/A265080/b265080.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A265080 Daniel R. L. Brown, <a href="https://ia.cr/2015/375">Bounds on surmising remixed keys</a>, IACR, Report 2015/375, 2015-2016. See Table 1.
%F A265080 T(n,k) = n! * [x^n] Sum_{j>=0} (exp(x)^k - (Sum_{i=0..j} x^i/i!)^k). - _Andrew Howroyd_, Aug 09 2025
%e A265080 Array begins:
%e A265080   0, 0,   0,   0,    0,    0, ...
%e A265080   0, 1,   2,   3,    4,    5, ...
%e A265080   0, 2,   6,  12,   20,   30, ...
%e A265080   0, 3,  18,  51,  108,  195, ...
%e A265080   0, 4,  44, 192,  544, 1220, ...
%e A265080   0, 5, 110, 675, 2540, 7145, ...
%e A265080   ...
%o A265080 (PARI) Q(p)={my(S=Set(p));prod(i=1, #S, (#select(t->t==S[i],p))!)}
%o A265080 T(n,k)={my(s=0); if(n, forpart(p=n, s+=p[#p]*n!*(#p)!*binomial(k,#p) / (prod(i=1,#p,p[i]!) * Q(Vec(p))))); s} \\ _Andrew Howroyd_, Mar 20 2021
%o A265080 (PARI) T(n,k) = {n!*polcoef(sum(j=0, n, exp(x + O(x*x^n))^k - sum(i=0, j, x^i/i!, O(x*x^n))^k), n)} \\ _Andrew Howroyd_, Aug 09 2025
%Y A265080 Rows n=1..5 are A001477, A002378, A064043, A265081, A265082.
%Y A265080 Columns k=1..5 are A001477, A230137, A265083, A265084, A265085.
%Y A265080 Main diagonal is A208250.
%K A265080 nonn,tabl
%O A265080 0,8
%A A265080 _N. J. A. Sloane_, Jan 01 2016
%E A265080 More terms from _Henry Bottomley_, Mar 20 2021