A370262 -id:A370262 - cp's OEIS frontend

A371697 Row sums of triangle A370262.

1, 2, 11, 113, 1732, 35509, 914213, 28372686, 1031486867, 43009596241, 2023804597256, 106098215717113, 6133027601141401, 387563995785510010, 26581805841852520619, 1966679438751901515329, 156133093759166939659212, 13239463980230688450540781, 1194277034550296812912993853

Offset: 0

Peter Bala, Apr 03 2024

Cf. A370262.

seq(add(binomial(n+k, n-k)/(2*k+1) * (2*n+1)^k, k = 0..n), n = 0..20);

a(n) = Sum_{k = 0..n} binomial(n+k, n-k)/(2*k+1) * (2*n+1)^k

a(n)^2 = 2/(2*n + 1)^3 * (T(2*n+1, n+3/2) - 1), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.

A370260 a(n) = sqrt(A370259(2*n+1)).

Original entry on oeis.org

1, 3, 31, 617, 18529, 748859, 38149567, 2348482961, 169641143873, 14071599763379, 1318414335714015, 137720427724123513, 15871136311527376801, 2000355821099358166891, 273735526097742996298111, 40419227378551955037029921, 6405616571975691389276400257

Offset: 0

Peter Bala, Mar 11 2024

The sequence is conjectured to be integral [added 03 Mar 2024: now confirmed - see the Formula section].

Paolo Xausa, Table of n, a(n) for n = 0..300

Cf. A008310, A342205, A370259, A370261, A370262.

A370259 := n -> simplify( (ChebyshevT(n, n+1) - 1)/n^3 ):
seq(sqrt(A370259(2*n+1)), n = 0..20);

Table[Sqrt[(ChebyshevT[k, k + 1] - 1)/k^3], {k, 1, 40, 2}] (* Paolo Xausa, Jul 24 2024 *)

a(n) = sqrt( (T(2*n+1, 2*n+2) - 1)/(2*n+1)^3 ), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
a(n) = sqrt( Sum_{k = 1..2*n+1} (2^k)*(2*n + 1)^(k-2)*binomial(2*n + k + 1, 2*k)/(2*n + k + 1) ).
a(n) = Sum_{k = 0..n} binomial(n+k, n-k)/(2*k + 1) * (4*n + 2)^k (shows the sequence to be integral) = R(n,2), where R(n, x) is the n-th row polynomial of A370262. - Peter Bala, Apr 03 2024

cp's OEIS Frontend

A371697 Row sums of triangle A370262.

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