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%I #26 Oct 15 2022 16:35:42
%S 1,1,1,1,1,1,1,1,2,1,1,1,3,4,1,1,1,4,7,8,1,1,1,5,10,17,16,1,1,1,6,13,
%T 28,41,32,1,1,1,7,16,41,76,99,64,1,1,1,8,19,56,121,208,239,128,1,1,1,
%U 9,22,73,176,365,568,577,256,1,1,1,10,25,92,241,576,1093,1552,1393,512,1
%N Square array whose rows have e.g.f. exp(x)*cosh(sqrt(k)*x), k>=0, read by ascending antidiagonals.
%C Rows are the binomial transforms of expansions of cosh(sqrt(k)*x), k >= 0.
%H G. C. Greubel, <a href="/A084097/b084097.txt">Antidiagonals n = 0..50, flattened</a>
%F From _Robert G. Wilson v_, Jan 02 2013: (Start)
%F A(n, k) = (1/2)*( (1 + sqrt(n))^k + (1 - sqrt(n))^k ) (array).
%F T(n, k) = A(n-k, k). (End)
%F T(n, k) = Sum_{j=0..floor(k/2)} binomial(k-j, j)*((n-k-1)/4)^j/(k-j), with T(n, 0) = 1 (antidiagonal triangle T(n,k)). - _G. C. Greubel_, Oct 15 2022
%e Array, A(n,k), begins:
%e .n\k.........0..1...2...3....4.....5......6......7.......8........9.......10
%e .0: A000012..1..1...1...1....1.....1......1......1.......1........1........1
%e .1: A000079..1..1...2...4....8....16.....32.....64.....128......256......512
%e .2: A001333..1..1...3...7...17....41.....99....239.....577.....1393.....3363
%e .3: A026150..1..1...4..10...28....76....208....568....1552.....4240....11584
%e .4: A046717..1..1...5..13...41...121....365...1093....3281.....9841....29525
%e .5: A084057..1..1...6..16...56...176....576...1856....6016....19456....62976
%e .6: A002533..1..1...7..19...73...241....847...2899...10033....34561...119287
%e .7: A083098..1..1...8..22...92...316...1184...4264...15632....56848...207488
%e .8: A084058..1..1...9..25..113...401...1593...5993...23137....88225...338409
%e .9: A003665..1..1..10..28..136...496...2080...8128...32896...130816...524800
%e 10: A002535..1..1..11..31..161...601...2651..10711...45281...186961...781451
%e 11: A133294..1..1..12..34..188...716...3312..13784...60688...259216..1125312
%e 12: A090042..1..1..13..37..217...841...4069..17389...79537...350353..1575613
%e 13: A125816..1..1..14..40..248...976...4928..21568..102272...463360..2153984
%e 14: A133343..1..1..15..43..281..1121...5895..26363..129361...601441..2884575
%e 15: A133345..1..1..16..46..316..1276...6976..31816..161296...768016..3794176
%e 16: A120612..1..1..17..49..353..1441...8177..37969..198593...966721..4912337
%e 17: A133356..1..1..18..52..392..1616...9504..44864..241792..1201408..6271488
%e 18: A125818..1..1..19..55..433..1801..10963..52543..291457..1476145..7907059
%e 25: A083578
%e - _Robert G. Wilson v_, Jan 02 2013
%e Antidiagonal triangle, T(n,k), begins:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 1, 2, 1;
%e 1, 1, 3, 4, 1;
%e 1, 1, 4, 7, 8, 1;
%e 1, 1, 5, 10, 17, 16, 1;
%e 1, 1, 6, 13, 28, 41, 32, 1;
%e 1, 1, 7, 16, 41, 76, 99, 64, 1;
%e 1, 1, 8, 19, 56, 121, 208, 239, 128, 1;
%e 1, 1, 9, 22, 73, 176, 365, 568, 577, 256, 1;
%e 1, 1, 10, 25, 92, 241, 576, 1093, 1552, 1393, 512, 1;
%t T[j_, k_] := Expand[((1 + Sqrt[j])^k + (1 - Sqrt[j])^k)/2]; T[1, 0] = 1; Table[ T[j - k, k], {j, 0, 11}, {k, 0, j}] // Flatten (* _Robert G. Wilson v_, Jan 02 2013 *)
%o (Magma)
%o function A084097(n,k)
%o if k eq 0 then return 1;
%o else return k*2^(k-1)*(&+[ Binomial(k-j,j)*((n-k-1)/4)^j/(k-j): j in [0..Floor(k/2)]]);
%o end if; return A084097; end function;
%o [A084097(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 15 2022
%o (SageMath)
%o def A084097(n,k):
%o if (k==0): return 1
%o else: return k*2^(k-1)*sum( binomial(k-j,j)*((n-k-1)/4)^j/(k-j) for j in range( (k+2)//2 ) )
%o flatten([[A084097(n,k) for k in range(n+1)] for n in range(15)]) # _G. C. Greubel_, Oct 15 2022
%Y Cf A140895, A221131.
%Y Rows: A000012, A000079, A001333, A026150, A046717, A084057, A002533, A083098, A084058, A003665,
%Y Rows: A002535, A133294, A090042, A125816, A133343, A133345, A120612, A133356, A125818, A083578.
%Y Columns: A000012, A000012, A000027, A016777, A028884, A134593.
%Y Rows include A011782, A001333, A026150, A046717, A002533.
%K easy,nonn,tabl
%O 0,9
%A _Paul Barry_, May 11 2003
%E Edited by _N. J. A. Sloane_, Jul 14 2010
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