cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -2, 1, 1, -2, -3, 0, 1, 1, -3, -2, 11, 6, 1, 1, -4, 1, 28, 1, 0, 1, 1, -5, 6, 45, -74, -81, -20, 1, 1, -6, 13, 56, -255, -92, 141, 0, 1, 1, -7, 22, 55, -554, 477, 1324, 363, 70, 1, 1, -8, 33, 36, -959, 2376, 2689, -3656, -1791, 0, 1
Offset: 0

Views

Author

Seiichi Manyama, May 02 2019

Keywords

Comments

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) + k*x*y). - Seiichi Manyama, Jul 11 2020
More generally, column k is the diagonal of the rational function r / ((1-r*x)*(1-r*y) + r-1 + (k-r+1)*r*x*y) for any nonzero real number r. - Seiichi Manyama, Jul 22 2020

Examples

			Square array begins:
  1,   1,   1,    1,    1,    1,      1, ...
  1,   0,  -1,   -2,   -3,   -4,     -5, ...
  1,  -2,  -3,   -2,    1,    6,     13, ...
  1,   0,  11,   28,   45,   56,     55, ...
  1,   6,   1,  -74, -255, -554,   -959, ...
  1,   0, -81,  -92,  477, 2376,   6475, ...
  1, -20, 141, 1324, 2689, -804, -20195, ...

Links

Crossrefs

Columns k=2..4 give (-1)^n * A098332, A116091, (-1)^n * A098341.
Main diagonal gives A307885.
T(n,n-1) gives A335310.
Cf. A307883, A336179.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

Formula

T(n,k) is the coefficient of x^n in the expansion of (1 - (k-1)*x - k*x^2)^n.
T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} (-k-1)^(n-j) * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-1) * T(n-2,k).

A307862 Coefficient of x^n in (1 + x - n*x^2)^n.

Original entry on oeis.org

1, 1, -3, -17, 49, 651, -1259, -38023, 26433, 2969299, 2225101, -289389891, -692529551, 33718183045, 143578976997, -4559187616649, -29119975483135, 699788001188403, 6188699469443869, -119828491083854707, -1404529670244379599, 22563726025297759345, 341997845736800473397
Offset: 0

Views

Author

Seiichi Manyama, May 02 2019

Keywords

Comments

Also coefficient of x^n in the expansion of 1/sqrt(1 - 2*x + (1+4*n)*x^2).

Links

Crossrefs

Main diagonal of A307860.
Cf. A002426, A187018, A307885.

Programs

  • Maple
    A307862:= n -> simplify(hypergeom([-n/2, (1-n)/2], [1], -4*n));
seq(A307862(n), n = 0..30); # G. C. Greubel, May 31 2020
  • Mathematica
    a[n_]:= SeriesCoefficient[(1 +x -n*x^2)^n, {x,0,n}]; Table[a[n], {n,0,30}] (* G. C. Greubel, May 31 2020 *)
  • PARI
    {a(n) = polcoef((1+x-n*x^2)^n, n)}
  • PARI
    {a(n) = sum(k=0, n\2, (-n)^k*binomial(n, k)*binomial(n-k, k))}
  • PARI
    {a(n) = sum(k=0, n\2, (-n)^k*binomial(n, 2*k)*binomial(2*k, k))}
  • Sage
    [ hypergeometric([-n/2, (1-n)/2], [1], -4*n).simplify_hypergeometric() for n in (0..30)] # G. C. Greubel, May 31 2020

Formula

a(n) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,k) * binomial(n-k,k).
a(n) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,2*k) * binomial(2*k,k).
a(n) = n! * [x^n] exp(x) * BesselI(0,2*sqrt(-n)*x). - Ilya Gutkovskiy, May 31 2020
a(n) = Hypergeometric2F1(-n/2, (1-n)/2; 1; -4*n). - G. C. Greubel, May 31 2020

A335310 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * (-n)^(n-k).

Original entry on oeis.org

1, 1, -2, 11, -74, 477, -804, -84425, 3315334, -102211207, 3005297956, -88338323709, 2627003399164, -78764141488043, 2341929797646648, -66394419743289105, 1609460569459689286, -18001777147777896975, -1625299659961386724524, 196005371138608184827003
Offset: 0

Views

Author

Ilya Gutkovskiy, May 31 2020

Keywords

Links

Crossrefs

Cf. A098332, A116091, A126869, A307884, A307885, A331657, A335309.

Programs

  • Mathematica
    Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] (-n)^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[SeriesCoefficient[1/Sqrt[1 + 2 (n - 2) x + n^2 x^2], {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[(2 - n) x] BesselI[0, 2 Sqrt[1 - n] x], {x, 0, n}], {n, 0, 19}]
Table[Hypergeometric2F1[-n, -n, 1, 1 - n], {n, 0, 19}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)^2*(1-n)^k); \\ Michel Marcus, Jun 01 2020

Formula

a(n) = central coefficient of (1 - (n - 2)*x - (n - 1)*x^2)^n.
a(n) = [x^n] 1 / sqrt(1 + 2*(n - 2)*x + n^2*x^2).
a(n) = n! * [x^n] exp((2 - n)*x) * BesselI(0,2*sqrt(1 - n)*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * (1-n)^k.

A336180 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^3.

Original entry on oeis.org

1, 0, -11, 136, -639, -25624, 1133245, -27431424, 259448833, 17402599792, -1405909697499, 63884679938960, -1830503703899519, -5324845289379264, 5494299851213052685, -496909924804074650624, 30201149245542631276545, -1236819213672144144878752, 5410434345252588202534741
Offset: 0

Views

Author

Seiichi Manyama, Jul 10 2020

Keywords

Links

Crossrefs

Main diagonal of A336179.
Cf. A241247, A307885.

Programs

  • Maple
    a := n -> hypergeom([-n, -n, -n], [1, 1], n):
seq(simplify(a(n)), n=0..18); # Peter Luschny, Dec 22 2020
  • Mathematica
    Array[Function[n, 1 + Sum[(-n)^k Binomial[n, k]^3, {k, n}]], 19, 0] (* Jan Mangaldan, Jul 14 2020 *)
  • PARI
    {a(n) = sum(k=0, n, (-n)^k*binomial(n, k)^3)}

Formula

a(n) = hypergeom([-n, -n, -n], [1, 1], n). - Peter Luschny, Dec 22 2020

A336728 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-n)^(n-k) * binomial(n,k) * binomial(n,k-1) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, 9, -174, 2575, -38219, 588833, -9274418, 141253551, -1739881142, -753419447, 1379742127908, -83720072007585, 4059017293956301, -184613801568558975, 8254420480122200214, -369177108304219471457, 16608406418618863804990, -750673988803431836351799
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2020

Keywords

Links

Crossrefs

Main diagonal of A336727.
Cf. A242369, A307885, A336713, A336714.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(-n)^(n - k) * Binomial[n, k] * Binomial[n , k - 1], {k, 1, n}] / n; Array[a, 21, 0] (* Amiram Eldar, Aug 02 2020 *)
  • PARI
    {a(n) = if(n==0, 1, sum(k=1, n, (-n)^(n-k)*binomial(n, k)*binomial(n, k-1))/n)}
  • PARI
    {a(n) = sum(k=0, n, (-n)^k*(n+1)^(n-k)*binomial(n, k)*binomial(n+k, n)/(k+1))}

Formula

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

A330497 a(n) = n! * Sum_{k=0..n} (-1)^k * binomial(n,k) * n^(n - k) / k!.

Original entry on oeis.org

1, 0, 1, 26, 1089, 70124, 6495985, 821315214, 136115947009, 28651724077976, 7470040450004001, 2363470644596843330, 892244303052345224641, 396227360441775922668036, 204487588996059177697597969, 121370399839482643287189048374
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2019

Keywords

Crossrefs

Cf. A009940, A025166, A061711, A277423, A307885, A318224, A319392, A330260.

Programs

  • Magma
    [Factorial(n)*&+[(-1)^k*Binomial(n,k)*n^(n-k)/Factorial(k):k in [0..n]]:n in [0..15]]; // Marius A. Burtea, Dec 18 2019
  • Mathematica
    Join[{1}, Table[n! Sum[(-1)^k Binomial[n, k] n^(n - k)/k!, {k, 0, n}], {n, 1, 15}]]
Join[{1}, Table[n^n n! LaguerreL[n, 1/n], {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[-x/(1 - n x)]/(1 - n x), {x, 0, n}], {n, 0, 15}]

Formula

a(n) = n! * [x^n] exp(-x/(1 - n*x)) / (1 - n*x).
a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k)^2 * n^k * k!.
a(n) ~ sqrt(2*Pi) * BesselJ(0,2) * n^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Dec 18 2019

A383133 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(n*k,k) * n^k.

Original entry on oeis.org

1, 0, 17, 1889, 412225, 151448249, 84430503361, 66535567456546, 70456680210155009, 96530372235620300465, 166169585125820280654001, 351113456811120647774884511, 893491183170443755035588745153, 2695374684029443253628238600963667, 9511442599320236554084097413617603681
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2025

Keywords

Crossrefs

Cf. A307885, A331657, A340972, A383121, A383132.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[n k, k] n^k, {k, 0, n}], {n, 0, 14}]

Formula

a(n) = [x^n] ((1 + n*x)^n - x)^n.
a(n) ~ exp(n - 1/2) * n^(2*n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Apr 19 2025
Showing 1-7 of 7 results.

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