A084097 Square array whose rows have e.g.f. exp(x)*cosh(sqrt(k)*x), k>=0, read by ascending antidiagonals.
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, 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, 9, 22, 73, 176, 365, 568, 577, 256, 1, 1, 1, 10, 25, 92, 241, 576, 1093, 1552, 1393, 512, 1
Offset: 0
Examples
Array, A(n,k), begins: .n\k.........0..1...2...3....4.....5......6......7.......8........9.......10 .0: A000012..1..1...1...1....1.....1......1......1.......1........1........1 .1: A000079..1..1...2...4....8....16.....32.....64.....128......256......512 .2: A001333..1..1...3...7...17....41.....99....239.....577.....1393.....3363 .3: A026150..1..1...4..10...28....76....208....568....1552.....4240....11584 .4: A046717..1..1...5..13...41...121....365...1093....3281.....9841....29525 .5: A084057..1..1...6..16...56...176....576...1856....6016....19456....62976 .6: A002533..1..1...7..19...73...241....847...2899...10033....34561...119287 .7: A083098..1..1...8..22...92...316...1184...4264...15632....56848...207488 .8: A084058..1..1...9..25..113...401...1593...5993...23137....88225...338409 .9: A003665..1..1..10..28..136...496...2080...8128...32896...130816...524800 10: A002535..1..1..11..31..161...601...2651..10711...45281...186961...781451 11: A133294..1..1..12..34..188...716...3312..13784...60688...259216..1125312 12: A090042..1..1..13..37..217...841...4069..17389...79537...350353..1575613 13: A125816..1..1..14..40..248...976...4928..21568..102272...463360..2153984 14: A133343..1..1..15..43..281..1121...5895..26363..129361...601441..2884575 15: A133345..1..1..16..46..316..1276...6976..31816..161296...768016..3794176 16: A120612..1..1..17..49..353..1441...8177..37969..198593...966721..4912337 17: A133356..1..1..18..52..392..1616...9504..44864..241792..1201408..6271488 18: A125818..1..1..19..55..433..1801..10963..52543..291457..1476145..7907059 25: A083578 - _Robert G. Wilson v_, Jan 02 2013 Antidiagonal triangle, T(n,k), begins: 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, 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, 9, 22, 73, 176, 365, 568, 577, 256, 1; 1, 1, 10, 25, 92, 241, 576, 1093, 1552, 1393, 512, 1;
Links
- G. C. Greubel, Antidiagonals n = 0..50, flattened
Crossrefs
Programs
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Magma
function A084097(n,k) if k eq 0 then return 1; else return k*2^(k-1)*(&+[ Binomial(k-j,j)*((n-k-1)/4)^j/(k-j): j in [0..Floor(k/2)]]); end if; return A084097; end function; [A084097(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 15 2022
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Mathematica
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 *) -
SageMath
def A084097(n,k): if (k==0): return 1 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 ) ) flatten([[A084097(n,k) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Oct 15 2022
Formula
From Robert G. Wilson v, Jan 02 2013: (Start)
A(n, k) = (1/2)*( (1 + sqrt(n))^k + (1 - sqrt(n))^k ) (array).
T(n, k) = A(n-k, k). (End)
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
Extensions
Edited by N. J. A. Sloane, Jul 14 2010
Comments