A084097 - OEIS

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

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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, 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