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A320956 a(n) = A000110(n) * A000111(n). The exponential limit of sec + tan. Row sums of A373428.

%I A320956 #34 Jun 13 2024 04:57:47
%S A320956 1,1,2,10,75,832,12383,238544,5733900,167822592,5859172975,
%T A320956 240072637440,11388362495705,618357843791872,38057876106154882,
%U A320956 2632817442236631040,203225803724876875315,17390464322078045896704,1640312648221489789841119,169667967895669459925991424
%N A320956 a(n) = A000110(n) * A000111(n). The exponential limit of sec + tan. Row sums of A373428.
%C A320956 We say that the sequence S is the exponential limit of the function f relative to the kernel K if and only if the exponential generating functions
%C A320956   egf(n) = Sum_{k=0..n} K(n, k)*f(x*(n-k)) generate a family of sequences
%C A320956 T(n) = k -> (k!/n!)*[x^k] egf(n) which converge to S. Convergence here means that for every fixed k the terms T(n)(k) differ from S(k) only for finitely many indices.
%C A320956 The paradigmatic example is to set f(x) = exp(x), K(n, k) = !k*binomial(n, k) (!n is the subfactorial of n) and obtain for S the Bell numbers. This example is set forth in A320955.
%C A320956 Let D(f)(x) represent the derivative of f(x) with respect to x and (D^(n))(f) the n-th derivative of f. Then the exponential limit of f is B(n)*((D^(n))(f))(0) where B(n) is the n-th Bell number: ExpLim(f) = f(0), (D(f))(0), 2*((D^(2))(f))(0), 5*((D^(3))(f))(0), 15*((D^(4))(f))(0), 52*((D^(5))(f))(0), ... Since exp is a fixed point of D and exp(0) = 1 we have the identity ExpLim(exp)[n] = B(n). Similarly ExpLim(sin)[n] = B(n)*mod(n,2)*(-1)^binomial(n,2).
%C A320956 If we set f = sec + tan and K(n, k) = !k*binomial(n, k) the exponential limit is this sequence, a(n).
%H A320956 Peter Luschny, <a href="/A320956/b320956.txt">Table of n, a(n) for n = 0..296</a>
%e A320956 Illustration of the convergence:
%e A320956   [0] 1, 0, 0,  0,  0,   0,     0,      0,       0, ... A000007
%e A320956   [1] 1, 1, 1,  2,  5,  16,    61,    272,    1385, ... A000111
%e A320956   [2] 1, 1, 2,  8, 40, 256,  1952,  17408,  177280, ... A000828
%e A320956   [3] 1, 1, 2, 10, 70, 656,  7442,  99280, 1515190, ... A320957
%e A320956   [4] 1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, ... A321394
%e A320956   [5] 1, 1, 2, 10, 75, 832, 12322, 232560, 5325325, ...
%e A320956   [6] 1, 1, 2, 10, 75, 832, 12383, 238272, 5693735, ...
%e A320956   [7] 1, 1, 2, 10, 75, 832, 12383, 238544, 5732515, ...
%e A320956   [8] 1, 1, 2, 10, 75, 832, 12383, 238544, 5733900, ...
%p A320956 ExpLim := proc(len, f) local kernel, sf, egf:
%p A320956 sf := proc(n) option remember; `if`(n <= 1, 1 - n, (n-1)*(sf(n-1) + sf(n-2))) end:
%p A320956 kernel := proc(n, k) option remember; binomial(n, k)*sf(k) end:
%p A320956 egf := n -> add(kernel(n, k)*f(x*(n-k)), k=0..n):
%p A320956 series(egf(len), x, len+2): seq(coeff(%, x, k)*k!/len!, k=0..len) end:
%p A320956 ExpLim(19, sec + tan);
%p A320956 # Alternative:
%p A320956 explim := (len, f) -> seq(combinat:-bell(n)*((D@@n)(f))(0), n=0..len):
%p A320956 explim(19, sec + tan);
%p A320956 # Or:
%p A320956 a := n -> A000110(n)*A000111(n): seq(a(n), n = 0..19);  # _Peter Luschny_, Jun 07 2024
%t A320956 m = 20; CoefficientList[Sec[x] + Tan[x] + O[x]^m, x] * Range[0, m-1]! *
%t A320956 BellB[Range[0, m-1]] (* _Jean-François Alcover_, Jun 19 2019 *)
%Y A320956 Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), A321394 (n=4).
%Y A320956 Cf. A320955 (exp), A320962 (log(x+1)), this sequence (sec+tan), A320958 (arcsin), A320959 (arctanh).
%Y A320956 Cf. A373428.
%K A320956 nonn
%O A320956 0,3
%A A320956 _Peter Luschny_, Nov 07 2018
%E A320956 Name extended by _Peter Luschny_, Jun 07 2024