A229580 -id:A229580 - cp's OEIS frontend

A368846 Triangle read by rows: T(n, k) = (-1)^(n + k)2binomial(2k - 1, n) binomial(2n + 1, 2k) for k > 0, and k^n for k = 0.

1, 0, 6, 0, 0, 30, 0, 0, -70, 140, 0, 0, 0, -840, 630, 0, 0, 0, 924, -6930, 2772, 0, 0, 0, 0, 18018, -48048, 12012, 0, 0, 0, 0, -12870, 216216, -300300, 51480, 0, 0, 0, 0, 0, -350064, 2042040, -1750320, 218790, 0, 0, 0, 0, 0, 184756, -5542680, 16628040, -9699690, 923780

Offset: 0

Peter Luschny, Jan 07 2024

The row sums of the inverse triangle (A368847/A368848) are the unsigned Bernoulli numbers |B(2n)|. To get the signed Bernoulli numbers B(2n), one only needs to change the sign factor in the definition from (-1)^(n + k) to (-1)^(n + 1).

Conjecture: |Sum_{j=0..k} T(k + j, k)| = A229580(k + 1) for k >= 0.

			[0] [1]
[1] [0, 6]
[2] [0, 0,  30]
[3] [0, 0, -70,  140]
[4] [0, 0,   0, -840,    630]
[5] [0, 0,   0,  924,  -6930,   2772]
[6] [0, 0,   0,    0,  18018,  -48048,   12012]
[7] [0, 0,   0,    0, -12870,  216216, -300300,    51480]
[8] [0, 0,   0,    0,      0, -350064, 2042040, -1750320, 218790]

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
Thomas Curtright, Scale Invariant Scattering and the Bernoulli Numbers, arXiv:2401.00586 [math-ph], Jan 2024.
Thomas L. Curtright, Bernoulli Partitions, arXiv:2502.09633 [math.CO], 2025.

Cf. A368847/A368848 (inverse), A369134, A369135, A002457 (main diagonal), A000367/A002445 (Bernoulli(2n)), A229580.

A368846[n_,k_] := If[k==0, Boole[n==0], (-1)^(n+k) 2 Binomial[2k-1, n] Binomial[2n+1, 2k]];
Table[A368846[n, k], {n,0,10}, {k,0,n}] (* Paolo Xausa, Jan 08 2024 *)

def A368846(n, k):
    if k == 0: return k^n
    if k  > n: return 0
    return (-1)^(n + k)*2*binomial(2*k - 1, n)*binomial(2*n + 1, 2*k)
for n in range(10): print([A368846(n, k) for k in range(n+1)])

A385601 Expansion of e.g.f. cosh(x)^2*(x+x^2/2).

Original entry on oeis.org

0, 1, 1, 6, 12, 40, 120, 224, 896, 1152, 5760, 5632, 33792, 26624, 186368, 122880, 983040, 557056, 5013504, 2490368, 24903680, 11010048, 121110528, 48234496, 578813952, 209715200, 2726297600, 905969664, 12683575296, 3892314112, 58384711680, 16642998272, 266287972352, 70866960384

Offset: 0

Enrique Navarrete, Jul 04 2025

a(n) is the number of ternary strings of length n that contain one or two 1's, an even number of 0's, and an even number of 2's.

A229580 and (1/3)*A229679 interleaved.

			a(5) = 40 since the strings are the 5 permutations of 10000, the 5 permutations of 12222 and the 30 permutations of 10022.
a(6) = 120 since the strings are the 15 permutations of 110000, the 90 permutations of 110022, and the 15 permutations of 112222.

Index entries for linear recurrences with constant coefficients, signature (0,12,0,-48,0,64).

Cf. A229580, A229679, A385083.

a(n) = n*2^(n-2) for n odd, n >= 3.
a(n) = binomial(n,2)*2^(n-3) for n even, n >= 4.
G.f.: x*(1 + x - 6*x^2 + 16*x^4 + 24*x^5 - 32*x^6 - 32*x^7)/(1 - 4*x^2)^3. - Stefano Spezia, Jul 04 2025

A386227 Expansion of e.g.f. cosh(x)^2*(1 + x + x^2/2).

Original entry on oeis.org

1, 1, 3, 6, 20, 40, 152, 224, 1024, 1152, 6272, 5632, 35840, 26624, 194560, 122880, 1015808, 557056, 5144576, 2490368, 25427968, 11010048, 123207680, 48234496, 587202560, 209715200, 2759852032, 905969664, 12817793024, 3892314112, 58921582592, 16642998272, 268435456000, 70866960384

Offset: 0

Enrique Navarrete, Jul 15 2025

a(n) is the number of ternary strings of length n that contain at most two 1's, an even number of 0's, and an even number of 2's.

			a(5) = 40 since the strings are the 30 permutations of 10022, the 5 permutations of 10000, and the 5 permutations of 12222.
a(6) = 152 since the strings are (number of permutations in parentheses): 110022 (90), 110000 (15), 112222 (15), 000022(15), 002222 (15), 222222 (1), 000000 (1).

Index entries for linear recurrences with constant coefficients, signature (0,12,0,-48,0,64).

Cf. A229580, A385083, A385601.

a(n) = n*2^(n-2) for odd n >= 3.
a(n) = 2^(n-3)*(binomial(n,2) + 4) for even n >= 4.
a(2n+1) = A229580(n+1).
G.f.: (1 + x - 9*x^2 - 6*x^3 + 32*x^4 + 16*x^5 - 8*x^6 - 32*x^7 - 32*x^8)/((1 - 2*x)^3*(1 + 2*x)^3).

A383608 Expansion of e.g.f. (1+x)*cosh(x)^2.

Original entry on oeis.org

1, 1, 2, 6, 8, 40, 32, 224, 128, 1152, 512, 5632, 2048, 26624, 8192, 122880, 32768, 557056, 131072, 2490368, 524288, 11010048, 2097152, 48234496, 8388608, 209715200, 33554432, 905969664, 134217728, 3892314112, 536870912, 16642998272, 2147483648, 70866960384, 8589934592

Offset: 0

Enrique Navarrete, May 01 2025

a(n) is the number of ternary strings of length n with an even number of 0s and 2s and one or no 1s.

A081294 and A229580 interleaved.

			a(5)=40 since the strings are the 30 permutations of 00221, the 5 permutations of 00001, and the 5 permutations of 22221.

Index entries for linear recurrences with constant coefficients, signature (0,8,0,-16).

Cf. A081294, A229580.

a(n)=2^(n-1) for n even, n > 0, a(0)=1.
a(n)=n*2^(n-2) for n odd, n > 1, a(1)=1.
G.f.: (1 + x - 6*x^2 - 2*x^3 + 8*x^4 + 8*x^5)/(1 - 4*x^2)^2. - Stefano Spezia, May 02 2025
a(n) = 8*a(n-2) - 16*a(n-4), n > 5.

cp's OEIS Frontend

A368846 Triangle read by rows: T(n, k) = (-1)^(n + k)2binomial(2k - 1, n) binomial(2n + 1, 2k) for k > 0, and k^n for k = 0.

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A385601 Expansion of e.g.f. cosh(x)^2*(x+x^2/2).

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A386227 Expansion of e.g.f. cosh(x)^2*(1 + x + x^2/2).

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A383608 Expansion of e.g.f. (1+x)*cosh(x)^2.

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cp's OEIS Frontend

A368846 Triangle read by rows: T(n, k) = (-1)^(n + k)*2*binomial(2*k - 1, n)* binomial(2*n + 1, 2*k) for k > 0, and k^n for k = 0.

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A385601 Expansion of e.g.f. cosh(x)^2*(x+x^2/2).

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A386227 Expansion of e.g.f. cosh(x)^2*(1 + x + x^2/2).

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A383608 Expansion of e.g.f. (1+x)*cosh(x)^2.

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A368846 Triangle read by rows: T(n, k) = (-1)^(n + k)2binomial(2k - 1, n) binomial(2n + 1, 2k) for k > 0, and k^n for k = 0.