A377664
a(n) = Sum_{j=0..n} binomial(n, j)*Euler(j, 0)*(2*n)^j. Main diagonal of A377666.
Original entry on oeis.org
1, 0, -3, 46, 497, -47524, -737891, 218380506, 4534099905, -3027853088648, -79034002960099, 99913537539058310, 3145444161956190577, -6725392006687056786732, -248035037340684934103427, 829076907459643714597871026, 35061737998144136797680434945, -172868475620109085260017037166096
Offset: 0
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a := n -> local j; add(binomial(n, j)*euler(j, 0)*(2*n)^j, j = 0..n):
seq(a(n), n = 0..16);
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a[0] := 1; a[n_] := Sum[Binomial[n, j] EulerE[j, 0] (2n)^j, {j, 0, n}];
Table[a[n], {n, 0, 16}]
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from math import comb
from sympy import euler
def A377664(n): return sum(comb(n,j)*euler(j,0)*(n<<1)**j for j in range(n+1)) # Chai Wah Wu, Nov 14 2024
A377665
a(n) = Sum_{j=0..n} binomial(n, j) * Euler(j, 0) * 10^j. Row 5 of A377666.
Original entry on oeis.org
1, -4, -9, 236, 981, -47524, -295029, 20208716, 167213961, -14741279044, -152462570049, 16429489441196, 203906790454941, -25968596099278564, -376012858170009069, 55254540434093713676, 914353480122881739921, -152277985980992039230084, -2834887281233334168196089
Offset: 0
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a := n -> local j; add(binomial(n, j)*euler(j, 0)*10^j, j = 0..n):
# Alternative:
a := n -> add(10^j*(1-2^j)*bernoulli(j)*binomial(n+1, j), j = 0..n+1) / (5*(n+1)):
seq(a(n), n = 0..18);
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a[n_] := 5^n (4^(n+1) HurwitzZeta[-n, 1/(20)] - 2^(n + 1) HurwitzZeta[-n, 1/(10)]);
Table[Round[N[a[n], 64]], {n, 0, 18}]
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from mpmath import *
mp.dps = 32; mp.pretty = True
def a(n): return int(imag(2*I*(1+sum(binomial(n, j)*polylog(-j, I)*5^j for j in range(n+1)))))
print([a(n) for n in range(19)])
A378066
Array read by ascending antidiagonals: A(n, k) = (-2*n)^k * Euler(k, (n - 1)/(2*n)) for n >= 1 and A(0, k) = 1.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -3, -2, 1, 1, 1, -8, -11, 0, 1, 1, 1, -15, -26, 57, 16, 1, 1, 1, -24, -47, 352, 361, 0, 1, 1, 1, -35, -74, 1185, 1936, -2763, -272, 1, 1, 1, -48, -107, 2976, 6241, -38528, -24611, 0, 1
Offset: 0
Array starts:
[0] 1, 1, 1, 1, 1, 1, 1, ... A000012
[1] 1, 1, 0, -2, 0, 16, 0, ... A155585
[2] 1, 1, -3, -11, 57, 361, -2763, ... A188458
[3] 1, 1, -8, -26, 352, 1936, -38528, ... A000810
[4] 1, 1, -15, -47, 1185, 6241, -230895, ... A000813
[5] 1, 1, -24, -74, 2976, 15376, -906624, ... A378065
[6] 1, 1, -35, -107, 6265, 32041, -2749355, ...
[7] 1, 1, -48, -146, 11712, 59536, -6997248, ...
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A := (n, k) -> ifelse(n = 0, 1, (-2*n)^k * euler(k, (n - 1) / (2*n))):
for n from 0 to 7 do seq(A(n, k), k = 0..9) od; # row by row
# Alternative:
A := proc(n, k) local j; add(binomial(k, j)*euler(j, 1/2)*(-2*n)^j, j = 0..k) end: seq(seq(A(n - k, k), k = 0..n), n = 0..10);
# Using generating functions:
egf := n -> exp(x)/cosh(n*x): ser := n -> series(egf(n), x, 14):
row := n -> local k; seq(k!*coeff(ser(n), x, k), k = 0..7):
seq(lprint(row(n)), n = 0..7);
A377663
a(n) = 2*n^3 - 3*n + 1.
Original entry on oeis.org
1, 0, 11, 46, 117, 236, 415, 666, 1001, 1432, 1971, 2630, 3421, 4356, 5447, 6706, 8145, 9776, 11611, 13662, 15941, 18460, 21231, 24266, 27577, 31176, 35075, 39286, 43821, 48692, 53911, 59490, 65441, 71776, 78507, 85646, 93205, 101196, 109631, 118522, 127881, 137720
Offset: 0
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[2*n^3 - 3*n + 1 : n in [0..60]]; // Wesley Ivan Hurt, Aug 05 2025
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a := n -> 2*n^3 - 3*n + 1: seq(a(n), n = 0..41);
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LinearRecurrence[{4,-6,4,-1},{1, 0, 11, 46},42] (* James C. McMahon, Nov 14 2024 *)
Showing 1-4 of 4 results.
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