A387430 -id:A387430 - cp's OEIS frontend

A386621 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4kx - 4*x^2).

1, 1, 0, 1, 2, 2, 1, 4, 8, 0, 1, 6, 26, 32, 6, 1, 8, 56, 184, 136, 0, 1, 10, 98, 576, 1366, 592, 20, 1, 12, 152, 1328, 6216, 10424, 2624, 0, 1, 14, 218, 2560, 18886, 68976, 80996, 11776, 70, 1, 16, 296, 4392, 45256, 276208, 779456, 637424, 53344, 0

Offset: 0

Seiichi Manyama, Aug 29 2025

			Square array begins:
   1,    1,     1,      1,       1,        1,        1, ...
   0,    2,     4,      6,       8,       10,       12, ...
   2,    8,    26,     56,      98,      152,      218, ...
   0,   32,   184,    576,    1328,     2560,     4392, ...
   6,  136,  1366,   6216,   18886,    45256,    92886, ...
   0,  592, 10424,  68976,  276208,   822800,  2020392, ...
  20, 2624, 80996, 779456, 4114004, 15235520, 44758244, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A126869, A006139, A098443, A387428.

Main diagonal gives A387430.

a(n, k) = sum(j=0, n\2, (k^2+1)^j*(2*k)^(n-2*j)*binomial(n, 2*j)*binomial(2*j, j));

A(n,k) = Sum_{j=0..n} (k-i)^j * (k+i)^(n-j) * binomial(n,j)^2, where i is the imaginary unit.
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-2*j) * binomial(2*(n-j),n-j) * binomial(n-j,j).
n*A(n,k) = 2*k*(2*n-1)*A(n-1,k) + 4*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} (k^2+1)^j * (2*k)^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
A(n,k) = [x^n] (1 + 2*k*x + (k^2+1)*x^2)^n.
E.g.f. of column k: exp(2*k*x) * BesselI(0, 2*sqrt(k^2+1)*x).

A387459 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k), where i is the imaginary unit.

Original entry on oeis.org

1, 2, 11, 96, 1121, 16280, 281987, 5666304, 129488641, 3315041568, 93958705499, 2920298135040, 98749216968481, 3608920706225536, 141743544911838547, 5953777300691189760, 266315973364196014081, 12638365012375994704384, 634207216217264733599531, 33552879853099295377612800

Offset: 0

Vaclav Kotesovec, Aug 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Cf. A387430.

C := ComplexField(); [Floor(Abs( ((1 + I*n)*(-I + n)^n + (1 - I*n)*(I + n)^n)/2)): n in [0..30]]; // Vincenzo Librandi, Aug 30 2025

Mathematica

Table[Sum[(n-I)^k*(n+I)^(n-k), {k, 0, n}], {n, 0, 20}] (* or *) Table[((1 + I*n)*(-I + n)^n + (1 - I*n)*(I + n)^n)/2, {n, 0, 20}]

PARI

a(n) = sum(k=0, n, (n-I)^k * (n+I)^(n-k)); \\ Michel Marcus, Aug 30 2025

Formula

a(n) = ((1 + i*n)*(-i + n)^n + (1 - i*n)*(i + n)^n)/2, where i is the imaginary unit.

For n > 0, a(n) = (1 + n^2)^(n/2) * (cos(n*arctan(1/n)) + n*sin(n*arctan(1/n))).

a(n) ~ sin(1) * n^(n+1).

A387467 a(n) = Sum_{k=0..n} (1-ni)^k (1+ni)^(n-k) binomial(n,k)^2, where i is the imaginary unit.

Original entry on oeis.org

1, 2, 14, 128, 2566, 44752, 1523724, 39267328, 1893328966, 64541150912, 4029767542756, 170848520912896, 13100724115628956, 664175960969073152, 60396776494002647768, 3563049510869692907520, 374818464874078558810694, 25220474024437034383526912, 3012865557320147302034729844

Offset: 0

Seiichi Manyama, Aug 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Main diagonal of A387466.

Cf. A387430.

[(&+[n^(2*k) * Binomial(2*(n-k),n-k) * Binomial(n-k,k): k in [0..Floor(n/2)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[(n^2+1)^k*2^(n-2*k)*Binomial[n,2*k]*Binomial[2*k,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, Sep 01 2025 *)

a(n) = sum(k=0, n\2, (n^2+1)^k*2^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k));

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

cp's OEIS Frontend

A386621 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4kx - 4*x^2).

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Formula

A387459 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k), where i is the imaginary unit.

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Magma

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Formula

A387467 a(n) = Sum_{k=0..n} (1-ni)^k (1+ni)^(n-k) binomial(n,k)^2, where i is the imaginary unit.

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Magma

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cp's OEIS Frontend

A386621 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4*k*x - 4*x^2).

Views

Author

Keywords

Examples

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Programs

PARI

Formula

A387459 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k), where i is the imaginary unit.

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Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387467 a(n) = Sum_{k=0..n} (1-n*i)^k * (1+n*i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.

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Programs

Magma

Mathematica

PARI

Formula

A386621 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4kx - 4*x^2).

A387467 a(n) = Sum_{k=0..n} (1-ni)^k (1+ni)^(n-k) binomial(n,k)^2, where i is the imaginary unit.