A088874 - OEIS

A088874

T(n, k) = [x^k] (2*n)! [z^(2*n)] 1/cos(z)^x, triangle read by rows, for 0 <= k <= n.

%I #52 Mar 24 2020 06:52:10

%S 1,0,1,0,2,3,0,16,30,15,0,272,588,420,105,0,7936,18960,16380,6300,945,

%T 0,353792,911328,893640,429660,103950,10395,0,22368256,61152000,

%U 65825760,36636600,11351340,1891890,135135,0,1903757312

%N T(n, k) = [x^k] (2*n)! [z^(2*n)] 1/cos(z)^x, triangle read by rows, for 0 <= k <= n.

%C Previous name was: Triangle read by rows, given by [0, 2, 6, 12, 20, 30, 42, 56, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, ...] where Delta is the operator defined in A084938.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) #09.7.6.

%H Ghislain R. Franssens, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles </a>, JIS 9 (2006) #06.4.1.

%H Alan D. Sokal, <a href="https://arxiv.org/abs/1804.04498">The Euler and Springer numbers as moment sequences</a>, arXiv:1804.04498 [math.CO], 2018.

%F T(n, k) = A085734(n-1, k-1) for n>0 and k>0.

%F T(n, k) = [x^k] (2*n)! [z^(2*n)] sec(z)^x. - _Peter Luschny_, Jul 01 2019

%e Triangle starts:

%e [0] 1

%e [1] 0, 1

%e [2] 0, 2, 3

%e [3] 0, 16, 30, 15

%e [4] 0, 272, 588, 420, 105

%e [5] 0, 7936, 18960, 16380, 6300, 945

%e [6] 0, 353792, 911328, 893640, 429660, 103950, 10395

%e [7] 0, 22368256, 61152000, 65825760, 36636600, 11351340, 1891890, 135135

%p ser := series(sec(z)^x, z, 24): row := n -> n!*coeff(ser, z, n):

%p seq(seq(coeff(row(2*n), x, k), k=0..n), n=0..8); # _Peter Luschny_, Jul 01 2019

%t T[1, 1] = 1; T[n_, k_] := Sum[(1/2^(j-1))*StirlingS1[j, k-1]*Sum[(-1)^(i + k + n)*(i-j)^(2(n-1)) Binomial[2j, i], {i, 0, j-1}]/j!, {j, 1, n-1}];

%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 14 2018, after _Vladimir Kruchinin_ *)

%t a[n_] := (2n)! SeriesCoefficient[Sec[z]^x, {z, 0, 2n}] // CoefficientList[#, x] &;

%t Table[a[n], {n, 0, 8}] // Flatten (* _Peter Luschny_, Jul 01 2019 *)

%o (Sage) # uses [A241171]

%o def fr2_row(n):

%o if n == 0: return [1]

%o S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))

%o L = expand(S).list()

%o return sum(L[k]*binomial(x+k, n) for k in (0..n-1)).list()

%o A088874_row = lambda n: [(-1)^(n-k)*m for k,m in enumerate(fr2_row(n))]

%o for n in (0..7): print(A088874_row(n)) # _Peter Luschny_, Sep 19 2017

%Y Another version of the triangle A085734. A signed version is A318146.

%Y Diagonals give: A000007 A000182 A001147, row sums A000364.

%K nonn,tabl,easy

%O 0,5

%A _Philippe Deléham_, Nov 26 2003

%E New name by _Peter Luschny_, Jul 01 2019