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.

User: Vaclav Kotesovec

Vaclav Kotesovec's wiki page.

Vaclav Kotesovec has authored 2320 sequences. Here are the ten most recent ones:

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

Views

Author

Vaclav Kotesovec, Aug 29 2025

Keywords

Links

Crossrefs

Cf. A387430.

Programs

  • Magma
    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).

A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

Original entry on oeis.org

1, 0, 2, 24, 252, 2880, 38280, 594720, 10565520, 209502720, 4558407840, 107702179200, 2744400415680, 75016089308160, 2189152249764480, 67906418407027200, 2230160988344889600, 77271779968704921600, 2815893910009609228800, 107629691727791474841600, 4304364116456244429388800
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 24 2025

Keywords

Comments

In general, if s >= 1, 1 <= r <= s and e.g.f. = exp(x^r/(1-x)^s) then for n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + (s-r)*k - 1, s*k - 1)/k!.

Links

Crossrefs

Cf. A293012, A000262, A082579, A091695, A361283.
Cf. A052845, A052887, A386514.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x^2/(1-x)^4))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Aug 25 2025
  • Mathematica
    nmax=20; CoefficientList[Series[E^(x^2/(1-x)^4), {x, 0, nmax}], x] * Range[0, nmax]!
nmax=20; Join[{1}, Table[n!*Sum[Binomial[n+2*k-1, 4*k-1]/k!, {k, 1, n}], {n, 1, nmax}]]
Join[{1}, Table[n!*n*(n - 1)*(n + 1)/6 * HypergeometricPFQ[{1 - n/2, 3/2 - n/2, 1 + n/2, 3/2 + n/2}, {5/4, 3/2, 7/4, 2}, 1/16], {n, 1, 20}]]

Formula

For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + 2*k - 1, 4*k - 1)/k!.
a(n) = 5*(n-1)*a(n-1) - 2*(n-1)*(5*n-11)*a(n-2) + 2*(n-2)*(n-1)*(5*n-14)*a(n-3) - 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 2^(1/5) * 5^(-1/2) * exp(1/80 - 2^(-9/5)*n^(2/5)/3 + 5*2^(-8/5)*n^(4/5) - n) * n^(n - 1/10).

A385867 Permanent of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.

Original entry on oeis.org

1, 1, 10, 2568, 32455296, 33171803873280, 4092783209652289536000, 85191758794180067056209100800000, 398579307845175105508944536142159544320000000, 538664594626853888213693114387037430238145253736448000000000, 262763300482667111090711396658972748636113942776939213363557171200000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 08 2025

Keywords

Links

Crossrefs

Cf. A059332.

Programs

  • Mathematica
    Join[{1}, Table[Permanent[Table[(i + j - 1)!, {i, 1, n}, {j, 1, n}]], {n, 1, 10}]]
  • PARI
    a(n) = {matpermanent(matrix(n, n, i, j, (i + j - 1)!))};
for(n=0, 10, print1(a(n), ", "))
  • Python
    from math import factorial
from sympy import Matrix
def A385867(n): return Matrix(n, n, lambda i, j: factorial(i+j+1)).per() if n else 1 # Chai Wah Wu, Aug 09 2025

A386880 a(n) = [x^n] 1/(1 - x)^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 6, 56, 715, 11628, 230230, 5379616, 145008513, 4431613550, 151473214816, 5727160371180, 237377895350076, 10704005376506540, 521748877569771510, 27338999059076777600, 1532576541123942256285, 91527291781199227579626, 5801648509628587739612170, 389032765009190361630625600
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 06 2025

Keywords

Crossrefs

Cf. A014068, A054688, A116508, A177234, A386879.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1-x)^(n*(n+1)/2), {x, 0, n}], {n, 0, 25}]
Join[{1}, Table[Binomial[n*(n + 3)/2, n]*(n + 1)/(n + 3), {n, 1, 25}]]

Formula

a(n) ~ exp(n+2) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)).
For n > 0, a(n) = binomial(n*(n+3)/2, n) * (n+1)/(n+3).

A386879 a(n) = [x^n] 1/(1 - x)^(n*(n-1)/2).

Original entry on oeis.org

1, 0, 1, 10, 126, 2002, 38760, 888030, 23535820, 708930508, 23930713170, 895068996640, 36749279048405, 1643385429346680, 79515468511191440, 4139207762053520646, 230672804560960311000, 13703037308872895467960, 864424422377992704918690, 57711135174726478041405270, 4065392394346039279040037520
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 06 2025

Keywords

Crossrefs

Cf. A014068, A054688, A116508, A177234, A386880.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1-x)^(n*(n-1)/2), {x, 0, n}], {n, 0, 25}]
Join[{1}, Table[Binomial[n*(n+1)/2, n] * (n-1) / (n+1), {n, 1, 25}]]

Formula

a(n) ~ exp(n) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)).
For n > 0, a(n) = binomial(n*(n+1)/2, n) * (n-1)/(n+1).

A386813 Coefficients in q-expansion of E_2^3 * E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 408, 28872, -2685984, 24039336, 776610576, -657274464, -112765274688, -1315204139160, -9184174537416, -47705529895632, -201727238619744, -730623451715808, -2340991131399984, -6787572064867008, -18105120840067776, -44991518932447512, -105189400371536208, -233200610257765464
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A004009, A386781, A386785.

Programs

  • Mathematica
    terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^3*E4[x]^2, {x, 0, terms}], x]

A386814 Coefficients in q-expansion of E_2^4 * E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Original entry on oeis.org

1, -600, 34920, -157920, -23913240, 297457776, 3581091360, -13666238400, -458367407640, -4230394757880, -25457298127632, -118465178148000, -459399324219360, -1550209298287440, -4682236500918720, -12910757263315776, -32979872278342680, -78921341989665840, -178491991660958520
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A013973, A386785.

Programs

  • Mathematica
    terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^4*E6[x], {x, 0, terms}], x]

A386815 Coefficients in q-expansion of E_2^4 * E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 384, 19008, -3408384, 86384832, 390216960, -20773815552, -154767455232, 1360271378880, 30429758560128, 278226995437440, 1749537534970368, 8664534035259648, 36062711146189056, 131104383085776384, 427185615341306880, 1270776436150340544, 3499300888293305088, 9016032242401655616
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A004009, A386787.

Programs

  • Mathematica
    terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^4*E4[x]^2, {x, 0, 20}], x]

A386816 Coefficients in q-expansion of E_2^2 * E_4^3, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 672, 145152, 8663424, -337036224, -6505531200, 40579467264, 1996981485312, 25931378854080, 210242562994464, 1273050737441280, 6245511315490944, 26057670474216192, 95466371280176064, 314217417062264832, 945050326572360960, 2631525623493208512, 6854684254893824832
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A004009, A386787.

Programs

  • Mathematica
    terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^2*E4[x]^3, {x, 0, 20}], x]

A386817 Coefficients in q-expansion of E_2^3 * E_4 * E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -336, -114912, 4151616, 100931712, -2848456800, -37865826432, 222362076288, 7928555745600, 86986313152368, 620751040620480, 3392046804500928, 15293330001535488, 59435665658243616, 204976008706800384, 640351567531186560, 1840291945275505344, 4923361835292283488
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A004009, A013973, A386787.

Programs

  • Mathematica
    terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^3*E4[x]*E6[x], {x, 0, 20}], x]

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