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A185951 Exponential Riordan array (1, x*cosh(x)).

%I A185951 #39 Feb 25 2025 09:00:14
%S A185951 1,0,1,3,0,1,0,12,0,1,5,0,30,0,1,0,120,0,60,0,1,7,0,735,0,105,0,1,0,
%T A185951 896,0,2800,0,168,0,1,9,0,15372,0,8190,0,252,0,1,0,5760,0,114240,0,
%U A185951 20160,0,360,0,1,11,0,270765,0,556710,0,43890,0,495,0,1,0,33792,0,4118400,0,2084544,0,87120,0,660,0,1
%N A185951 Exponential Riordan array (1, x*cosh(x)).
%C A185951 The column k=0 of the array (which contains T(0,0)=1 and otherwise zero) is not included in the triangle.
%C A185951 Also the Bell transform of the sequence "a(n) = n+1 if n is even else 0". For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016
%H A185951 G. C. Greubel, <a href="/A185951/b185951.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A185951 Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A185951 T(n,k) = binomial(n,k)/(2^k) * Sum_{i=0..k} binomial(k,i) *(k-2*i)^(n-k), n > k; T(n,n) = 1.
%e A185951 Array begins
%e A185951    1,
%e A185951    0,   1,
%e A185951    3,   0,    1,
%e A185951    0,  12,    0,    1,
%e A185951    5,   0,   30,    0,    1,
%e A185951    0,  120,   0,   60,    0,    1,
%e A185951    7,   0,   735,   0,   105,   0,    1,
%e A185951    0,  896,   0,  2800,   0,   168,   0,   1,
%e A185951    9,   0,  15372,  0,  8190,   0,   252,  0,   1,
%e A185951    0, 5760,   0, 114240,  0,  20160,  0,  360,  0,  1,
%e A185951   11,   0, 270765,  0, 556710,  0,  43890, 0,  495, 0,  1,
%e A185951    0, 33792,  0, 4118400, 0, 2084544, 0, 87120, 0, 660, 0, 1.
%p A185951 A185951 := proc(n,k)
%p A185951    if n =k then
%p A185951       1;
%p A185951    else
%p A185951       binomial(n,k)/2^k * add( binomial(k,i)*(k-2*i)^(n-k),i=0..k) ;
%p A185951    end if;
%p A185951 end proc: # _R. J. Mathar_, Feb 22 2011
%p A185951 # The function BellMatrix is defined in A264428.
%p A185951 # Adds (1,0,0,0, ..) as column 0.
%p A185951 BellMatrix(n -> `if`(n::even,n+1,0), 10); # _Peter Luschny_, Jan 29 2016
%t A185951 t[n_, k_] := Binomial[n, k]/(2^k)* Sum[ Binomial[k, i]*(k-2*i)^(n-k), {i, 0, k}]; t[n_, n_] = 1; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 14 2013, from formula *)
%t A185951 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A185951 B = BellMatrix[Function[n, If[EvenQ[n], n + 1, 0]], rows = 12];
%t A185951 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A185951 (PARI) T(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
%o A185951 row(n) = vector(n, k, T(n,k)); \\ _Michel Marcus_, Feb 25 2025
%Y A185951 Cf. A264428.
%K A185951 nonn,tabl
%O A185951 1,4
%A A185951 _Vladimir Kruchinin_, Feb 11 2011