A050158 -id:A050158 - cp's OEIS frontend

A296770 Row sums of A050158.

1, 5, 24, 111, 500, 2210, 9632, 41531, 177564, 754014, 3184016, 13382710, 56026984, 233765636, 972504704, 4035441491, 16707488684, 69033916166, 284733161264, 1172510645666, 4821324991064, 19799091571676, 81208982686784, 332726301861086, 1361862906980120

Offset: 0

Peter Luschny, Dec 22 2017

nonn

Cf. A002457, A008549, A050158.

A296770 := n -> add(binomial(2*n+1, n+1) - binomial(2*n+1, n-k-1), k=0..n):
seq(A296770(n), n=0..24);

a[n_] := 4^n ((2 (2 + n) Gamma[3/2 + n])/(Sqrt[Pi] Gamma[2 + n]) - 1);
Table[a[n], {n, 0, 24}]

a(n) = Sum_{k=0..n} (binomial(2*n+1, n+1) - binomial(2*n+1, n-k-1)).
a(n) = 4^n*((2*(n + 2)*Gamma(n + 3/2))/(sqrt(Pi)*Gamma(n + 2)) - 1).
a(n) = (n/2+1)*binomial(2*(n+1), n+1) - 4^n.
a(n) ~ 4^n*(2*sqrt(n/Pi) - 1).
a(n) = A002457(n) - A008549(n).

A296662 Table read by rows, the odd rows of A296664, T(n, k) for n >= 0 and 0 <= k <= 2n.

Original entry on oeis.org

1, 2, 3, 2, 5, 9, 10, 9, 5, 14, 28, 34, 35, 34, 28, 14, 42, 90, 117, 125, 126, 125, 117, 90, 42, 132, 297, 407, 451, 461, 462, 461, 451, 407, 297, 132, 429, 1001, 1430, 1638, 1703, 1715, 1716, 1715, 1703, 1638, 1430, 1001, 429

Offset: 0

Peter Luschny, Dec 20 2017

Let v be the characteristic function of 1 (A063524) and M(n) for n >= 0 the symmetric Toeplitz matrix generated by the initial segment of v, then row n is the diagonal next to the main diagonal of M(2n+1)^(2n+1).

			The triangle starts:
0: [  1]
1: [  2,   3,   2]
2: [  5,   9,  10,   9,   5]
3: [ 14,  28,  34,  35,  34,  28,  14]
4: [ 42,  90, 117, 125, 126, 125, 117,  90,  42]
5: [132, 297, 407, 451, 461, 462, 461, 451, 407, 297, 132]

Cf. A000108, A001700, A050158, A296664, A296666, A296769 (row sums).

v := n -> `if`(n=1, 1, 0):
B := n -> LinearAlgebra:-ToeplitzMatrix([seq(v(j),j=0..n)], symmetric):
seq(convert(ArrayTools:-Diagonal(B(2*n+1)^(2*n+1), 1),list), n=0..6);

v[n_] := If[n == 1, 1, 0];
m[n_] := MatrixPower[ToeplitzMatrix[Table[v[k], {k, 0, n}]], n];
d[n_] := Diagonal[m[2 n + 1], 1];
Table[d[n], {n, 0, 6}] // Flatten

def T(n, k):
    if k > n:
        b = binomial(2*n+1, k - n - 1)
    else:
        b = binomial(2*n+1, n - k - 1)
    return binomial(2*n+1, n+1) - b
for n in (0..6):
    print([T(n, k) for k in (0..2*n)])

T(n, n) = A001700(n).
T(n, 0) = T(n, 2*n) = A000108(n+1).
T(n, k) = binomial(2*n+1, n+1) - binomial(2*n+1, n-k-1) for k=0..n.
T(n, k) = binomial(2*n+1, n+1) - binomial(2*n+1, k-n-1) for k=n+1..2*n and n>0.

cp's OEIS Frontend

A296770 Row sums of A050158.

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