Original entry on oeis.org
1, 5, 161, 8749, 665857, 65160501, 7793761249, 1101696200669, 179689877047297, 33215554576822501, 6862186181491284001, 1566923219786361397005, 391868347839681254572801, 106523078497331434142611733, 31273034455313887578671676257
Offset: 0
-
{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(2*n+1)^(n-k)*(2*n)^k)}
-
{a(n) = polchebyshev(n, 1, 4*n+1)}
A322699
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is 1/2 * (-1 + Sum_{j=0..k} binomial(2*k,2*j)*(n+1)^(k-j)*n^j).
Original entry on oeis.org
0, 0, 0, 0, 1, 0, 0, 8, 2, 0, 0, 49, 24, 3, 0, 0, 288, 242, 48, 4, 0, 0, 1681, 2400, 675, 80, 5, 0, 0, 9800, 23762, 9408, 1444, 120, 6, 0, 0, 57121, 235224, 131043, 25920, 2645, 168, 7, 0, 0, 332928, 2328482, 1825200, 465124, 58080, 4374, 224, 8, 0
Offset: 0
Square array begins:
0, 0, 0, 0, 0, 0, 0, ...
0, 1, 8, 49, 288, 1681, 9800, ...
0, 2, 24, 242, 2400, 23762, 235224, ...
0, 3, 48, 675, 9408, 131043, 1825200, ...
0, 4, 80, 1444, 25920, 465124, 8346320, ...
0, 5, 120, 2645, 58080, 1275125, 27994680, ...
0, 6, 168, 4374, 113568, 2948406, 76545000, ...
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Unprotect[Power]; 0^0 := 1; Protect[Power]; Table[(-1 + Sum[Binomial[2 k, 2 j] (# + 1)^(k - j)*#^j, {j, 0, k}])/2 &[n - k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Michael De Vlieger, Jan 01 2019 *)
nmax = 9; row[n_] := LinearRecurrence[{4n+3, -4n-3, 1}, {0, n, 4n(n+1)}, nmax+1]; T = Array[row, nmax+1, 0]; A[n_, k_] := T[[n+1, k+1]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 06 2019 *)
-
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
(0..n).map{|i| (0..k).inject(-1){|s, j| s + ncr(2 * k, 2 * j) * (i + 1) ** (k - j) * i ** j} / 2}
end
def A322699(n)
a = []
(0..n).each{|i| a << A(i, n - i)}
ary = []
(0..n).each{|i|
(0..i).each{|j|
ary << a[i - j][j]
}
}
ary
end
p A322699(10)
A173174
a(n) = cosh(2*n*arcsinh(sqrt(n))).
Original entry on oeis.org
1, 3, 49, 1351, 51841, 2550251, 153090001, 10850138895, 886731088897, 82094249361619, 8491781781142001, 970614726270742103, 121485428812828080001, 16525390478051500325307, 2427469037137019032095121, 382956978214541873571486751, 64576903826545426454350012417, 11591229031806966336496244914595
Offset: 0
Cf.
A132592,
A146311 -
A146313,
A173115,
A173116 A173121,
A173127 -
A173131,
A173133,
A173134,
A173148,
A173151,
A173170,
A173171.
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[&+[Binomial(2*n, 2*k)*(n+1)^(n-k)*n^k: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Dec 29 2018
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A173174 := proc(n) cosh(2*n*arcsinh(sqrt(n))) ; expand(%) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011
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Table[Round[N[Cosh[(2 n) ArcSinh[Sqrt[n]]], 100]], {n, 0, 30}] (* Artur Jasinski *)
Join[{1}, a[n_]:=Sum[Binomial[2 n, 2 k] (n + 1)^(n - k) n^k, {k, 0, n}]; Array[a, 25]] (* Vincenzo Librandi, Dec 29 2018 *)
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{a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n+1)^(n-k)*n^k)} \\ Seiichi Manyama, Dec 26 2018
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{a(n) = polchebyshev(n, 1, 2*n+1)} \\ Seiichi Manyama, Dec 29 2018
Showing 1-3 of 3 results.