A362994
a(n) = lcm(1, 2, ..., n+1) * Bernoulli(n, 1). Alternating row sums of A362995 and A363154.
Original entry on oeis.org
1, 1, 1, 0, -2, 0, 10, 0, -84, 0, 2100, 0, -91212, 0, 420420, 0, -86894808, 0, 12796881240, 0, -123176186952, 0, 33154044803880, 0, -2317852458291480, 0, 114488177740536600, 0, -63580025062953158760, 0, 43435207772044760997000, 0, -2182849703429651931795120
Offset: 0
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A362994 := n -> ilcm(seq(i + 1, i = 0..n)) * bernoulli(n, 1):
seq(A362994(n), n = 0..32);
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A362994[n_]:=LCM@@Range[n+1]BernoulliB[n, 1];Array[A362994,50,0] (* Paolo Xausa, Aug 09 2023 *)
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a(n) = lcm([1..n+1])*subst(bernpol(n), 'x, 1); \\ Michel Marcus, Aug 09 2023
A362992
a(n) = (n + 1)^(n - 1) * lcm{k + 1 : 0 <= k <= n}. Main diagonal of triangle A362995.
Original entry on oeis.org
1, 2, 18, 192, 7500, 77760, 7058940, 220200960, 12053081880, 252000000000, 65362309994520, 1716349336289280, 645822919595173320, 20430218263561666560, 701330854833984375000, 51933349175015422033920, 35071094208630625451626320, 1487906280482935955379978240
Offset: 0
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A362992[n_]:=LCM@@Range[n+1](n+1)^(n-1);Array[A362992,20,0] (* Paolo Xausa, Aug 09 2023 *)
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a(n) = (n + 1)^(n - 1) * lcm(vector(n, k, k+1)); \\ Michel Marcus, May 20 2023
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def A362992(n: int) -> int:
return (n + 1)^(n - 1) * lcm(k + 1 for k in (0..n))
print([A362992(n) for n in (0..17)])
Original entry on oeis.org
1, 5, 139, 8920, 3140313, 386431752, 514497175965, 279843094350688, 309269596069652505, 148299194161602748000, 987531598214016848643771, 736658715557644007659575408, 8631341032315887149556551616665, 9247541236236479803404967113812944, 11617153578430950909883928682002963745
Offset: 0
A362996
Triangle read by rows. T(n, k) = numerator([x^k] R(n, n, x)), where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).
Original entry on oeis.org
1, 3, 1, 11, 14, 3, 25, 46, 117, 16, 137, 652, 3699, 1344, 125, 49, 568, 19197, 41728, 19375, 1296, 363, 9872, 621837, 2397184, 2084375, 334368, 16807, 761, 23664, 5338467, 17115136, 99109375, 7150032, 6705993, 262144
Offset: 0
The triangle T(n, k) begins:
[0] 1;
[1] 3, 1;
[2] 11, 14, 3;
[3] 25, 46, 117, 16;
[4] 137, 652, 3699, 1344, 125;
[5] 49, 568, 19197, 41728, 19375, 1296;
[6] 363, 9872, 621837, 2397184, 2084375, 334368, 16807;
[7] 761, 23664, 5338467, 17115136, 99109375, 7150032, 6705993, 262144;
.
The first few polynomials are:
[0] 1
[1] x + 3/2
[2] 3*x^2 + (14/3)*x + 11/6
[3] 16*x^3 + (117/4)*x^2 + (46/3)*x + 25/12
[4] 125*x^4 + (1344/5)*x^3 + (3699/20)*x^2 + (652/15)*x + 137/60
[5] 1296*x^5 + (19375/6)*x^4 + (41728/15)*x^3 + (19197/20)*x^2 + (568/5)*x + 49/20
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def R(n, k, x):
return add((1 / (u + 1)) * add(x^j * binomial(u, j) * (j + 1)^n
for j in (0..u)) for u in (0..k))
def A362996row(n: int) -> list[int]:
return [r.numerator() for r in R(n, n, x).list()]
for n in (0..7): print(A362996row(n))
A362997
Triangle read by rows. T(n, k) = denominator([x^k] R(n, n, x)), where R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).
Original entry on oeis.org
1, 2, 1, 6, 3, 1, 12, 3, 4, 1, 60, 15, 20, 5, 1, 20, 5, 20, 15, 6, 1, 140, 35, 140, 105, 42, 7, 1, 280, 35, 280, 105, 168, 7, 8, 1, 2520, 315, 280, 315, 504, 7, 72, 9, 1, 2520, 315, 280, 315, 504, 35, 360, 45, 10, 1, 27720, 3465, 3080, 3465, 5544, 385, 3960, 495, 110, 11, 1
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 2, 1;
[2] 6, 3, 1;
[3] 12, 3, 4, 1;
[4] 60, 15, 20, 5, 1;
[5] 20, 5, 20, 15, 6, 1;
[6] 140, 35, 140, 105, 42, 7, 1;
[7] 280, 35, 280, 105, 168, 7, 8, 1;
[8] 2520, 315, 280, 315, 504, 7, 72, 9, 1;
[9] 2520, 315, 280, 315, 504, 35, 360, 45, 10, 1;
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def R(n, k, x):
return add((1 / (u + 1)) * add(x^j * binomial(u, j) * (j + 1)^n
for j in (0..u)) for u in (0..k))
def A362997row(n: int) -> list[int]:
return [r.denominator() for r in R(n, n, x).list()]
for n in (0..9): print(A362997row(n))
Showing 1-5 of 5 results.