Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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