A305861 -id:A305861 - cp's OEIS frontend

A305862 a(n) = 3844^n - 5763^n + 220*2^n - 14.

14, 234, 1826, 10770, 55154, 260274, 1167026, 5059890, 21442994, 89438514, 368866226, 1509026610, 6137242034, 24853275954, 100327829426, 404059098930, 1624486948274, 6522713868594, 26165182536626, 104883769004850, 420204307937714, 1682825158192434, 6737324873467826

Offset: 0

Vincenzo Librandi, Jun 15 2018

From Bruno Berselli, Jun 15 2018: (Start)
a(0) = 2*7 and a(40) = 2*232110255958477539427146457 are semiprimes. For which values of n > 40 is a(n) semiprime?
For odd n, a(n) is divisible by 2*3.
For n == 3 (mod 4), a(n) is divisible by 2*3*5.
For n == 0 or 5 (mod 6), a(n) is divisible by 2*7.
For n == 2 or 4 (mod 5), a(n) is divisible by 2*11.
For n == 1 or 11 (mod 12), a(n) is divisible by 2*3*13.
For n == 15 (mod 16), a(n) is divisible by 2*3*5*17^2, etc.
If a(n) is divisible by 37 then it is also divisible by 3*5*7*13*19*73. (End)

Takao Komatsu, On poly-Euler numbers of the second kind, arXiv:1806.05515 [math.NT], 2018, page 11 (Lemma 3.4).
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A000453, A000918, A305861.

[384*4^n-576*3^n+220*2^n-14: n in [0..30]];

Table[384 4^n - 576 3^n + 220 2^n - 14, {n, 0, 30}]

a(n) = 384*4^n - 576*3^n + 220*2^n - 14; \\ Michel Marcus, Jul 03 2018

G.f.: 2*(7 + 47*x - 12*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)).
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4).
a(n) = 14*A000453(n+4) + 94*A000453(n+3) - 24*A000453(n+2) for n>1. - Bruno Berselli, Jun 15 2018

A305863 a(n) = 61445^n - 122884^n + 76163^n - 14722^n + 41.

Original entry on oeis.org

41, 1513, 19689, 175465, 1287657, 8420713, 51126249, 295141225, 1644285417, 8927926633, 47563308009, 249806529385, 1297882995177, 6687496584553, 34237868091369, 174415093507945, 885051189224937, 4477377106010473, 22596025278436329, 113818651291052905

Offset: 0

Vincenzo Librandi, Jul 04 2018

Takao Komatsu, On poly-Euler numbers of the second kind, arXiv:1806.05515 [math.NT], 2018, page 11 (Lemma 3.4).
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Cf. A007051, A081188, A305861, A305862.

[6144*5^n-12288*4^n+7616*3^n-1472*2^n+41: n in [0..20]];

Table[6144 5^n - 12288 4^n + 7616 3^n - 1472 2^n + 41, {n, 0, 30}]

G.f.: (41 + 898*x + 479*x^2 - 490*x^3 + 56*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)).

a(n) = 15*a(n-1) - 85*a(n-2) + 225*a(n-3) - 274*a(n-4) + 120*a(n-5) for n>5.

A316526 a(n) = 1228806^n - 3072005^n + 2649604^n - 902403^n + 9844*2^n - 122.

Original entry on oeis.org

122, 9966, 210134, 2741670, 27930182, 245220486, 1953210374, 14543545350, 103166087942, 706033804806, 4702595902214, 30675859444230, 196880387684102, 1247535454225926, 7825081688699654, 48684535015586310, 300917096071974662, 1850113238390115846

Offset: 0

Vincenzo Librandi, Jul 06 2018

Takao Komatsu, On poly-Euler numbers of the second kind, arXiv:1806.05515 [math.NT], 2018, page 11 (Lemma 3.4).
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A007051, A081188, A305861, A305862, A305863.

[122880*6^n-307200*5^n+264960*4^n-90240*3^n+9844*2^n-122: n in [0..20]];

Table[122880 6^n - 307200 5^n + 264960 4^n - 90240 3^n + 9844 2^n - 122, {n, 0, 20}]

G.f.: 2*(61 + 3702*x + 11099*x^2 - 8382*x^3 + 840*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)).

a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6) for n>6.

A345394 Array read by ascending antidiagonals: A(n, k) = n![x^n] Li(-k, 1 - exp(-4x))/(4*sinh(x)), where Li(n, z) is the polylogarithm function.

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 14, 37, 14, 1, 41, 234, 165, 30, 1, 122, 1513, 1826, 613, 62, 1, 365, 9966, 19689, 10770, 2085, 126, 1, 1094, 66637, 210134, 175465, 55154, 6757, 254, 1, 3281, 450834, 2236365, 2741670, 1287657, 260274, 21285, 510, 1, 9842, 3077713, 23819306, 41809933, 27930182, 8420713, 1167026, 65893, 1022, 1

Offset: 0

Stefano Spezia, Jun 17 2021

			n\k|   0     1      2       3        4 ...
---+----------------------------------
0  |   1     1      1       1        1 ...
1  |   2     6     14      30       62 ...
2  |   5    37    165     613     2085 ...
3  |  14   234   1826   10770    55154 ...
4  |  41  1513  19689  175465  1287657 ...
...

Beáta Bényi and Toshiki Matsusaka, Extensions of the combinatorics of poly-Bernoulli numbers, arXiv:2106.05585 [math.CO], 2021. See p. 9.
Komatsu, Takao Complementary Euler numbers. Period. Math. Hung. 75, No. 2, 302-314 (2017).
Takao Komatsu, On poly-Euler numbers of the second kind, arXiv:1806.05515 [math.NT], 2018.

Cf. A000012 (n = 0), A007051 (k = 0), A081188 (k = 1), A305861 (n = 2), A305862 (n = 3), A305863 (n = 4), A316526 (n = 5), A345393.

A[n_,k_]:=n!Coefficient[Series[PolyLog[-k,1-Exp[-4t]]/(4Sinh[t]),{t,0,n}],t,n]; Flatten[Table[A[n-k,k],{n,0,9},{k,0,n}]]

cp's OEIS Frontend

A305862 a(n) = 384*4^n - 576*3^n + 220*2^n - 14.

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A305863 a(n) = 6144*5^n - 12288*4^n + 7616*3^n - 1472*2^n + 41.

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A316526 a(n) = 122880*6^n - 307200*5^n + 264960*4^n - 90240*3^n + 9844*2^n - 122.

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A345394 Array read by ascending antidiagonals: A(n, k) = n!*[x^n] Li(-k, 1 - exp(-4*x))/(4*sinh(x)), where Li(n, z) is the polylogarithm function.

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A305862 a(n) = 3844^n - 5763^n + 220*2^n - 14.

A305863 a(n) = 61445^n - 122884^n + 76163^n - 14722^n + 41.

A316526 a(n) = 1228806^n - 3072005^n + 2649604^n - 902403^n + 9844*2^n - 122.

A345394 Array read by ascending antidiagonals: A(n, k) = n![x^n] Li(-k, 1 - exp(-4x))/(4*sinh(x)), where Li(n, z) is the polylogarithm function.