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

0,9

Comments

Rows are the binomial transforms of expansions of cosh(sqrt(k)*x), k >= 0.

Links

G. C. Greubel, Antidiagonals n = 0..50, flattened

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf A140895, A221131.

Rows: A000012, A000079, A001333, A026150, A046717, A084057, A002533, A083098, A084058, A003665,

Rows: A002535, A133294, A090042, A125816, A133343, A133345, A120612, A133356, A125818, A083578.

Columns: A000012, A000012, A000027, A016777, A028884, A134593.

Rows include A011782, A001333, A026150, A046717, A002533.

Sequence in context: A299037 A173072 A219272 * A306684 A293991 A288638

Adjacent sequences: A084094 A084095 A084096 * A084098 A084099 A084100

Keyword

easy,nonn,tabl

Author

Paul Barry, May 11 2003

Extensions

Edited by N. J. A. Sloane, Jul 14 2010

Status

approved