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 1-4 of 4 results.

A292975 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*(sec(x) + tan(x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 9, 9, 5, 1, 5, 16, 28, 24, 16, 1, 6, 25, 65, 93, 77, 61, 1, 7, 36, 126, 272, 338, 294, 272, 1, 8, 49, 217, 645, 1189, 1369, 1309, 1385, 1, 9, 64, 344, 1320, 3380, 5506, 6238, 6664, 7936, 1, 10, 81, 513, 2429, 8141, 18285, 27365, 31993, 38177, 50521
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 27 2017

Keywords

Comments

A(n,k) is the k-th binomial transform of A000111 evaluated at n.
Also column k is the boustrophedon transform of powers of k.

Examples

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k + 1)^2*x^2/2! + (k^3 + 3*k^2 + 3*k + 2)*x^3/3! + (k^4 + 4*k^3 + 6*k^2 + 8*k + 5)*x^4/4! + ...
Square array begins:
   1,   1,    1,     1,     1,     1,  ...
   1,   2,    3,     4,     5,     6,  ...
   1,   4,    9,    16,    25,    36,  ...
   2,   9,   28,    65,   126,   217,  ...
   5,  24,   93,   272,   645,  1320,  ...
  16,  77,  338,  1189,  3380,  8141,  ...

Links

Crossrefs

Columns k=0..2 give A000111, A000667, A000752.
Main diagonal gives A292976.

Programs

  • Maple
    b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
A:= proc(n, k) option remember; `if`(k=0, b(n, 0),
      add(binomial(n, j)*A(j, k-1), j=0..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 27 2017
  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[k x] (Sec[x] + Tan[x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: exp(k*x)*(sec(x) + tan(x)).

A296793 a(n) = n! * [x^n] exp(x)*(sec(x) + tan(x))^n.

Original entry on oeis.org

1, 2, 9, 67, 705, 9601, 160429, 3175579, 72638209, 1884974185, 54709142101, 1755923320559, 61748847320545, 2360991253910069, 97518218630249005, 4327060674324941491, 205272207854416078849, 10367500700785078039473, 555414837143457708584101, 31458118283019682610004279
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Links

Crossrefs

Cf. A000111, A000667, A292756, A292759, A292976.
Main diagonal of A322268.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[x] (Sec[x] + Tan[x])^n, {x, 0, n}], {n, 0, 19}]
  • PARI
    a(n) = Vec(serlaplace(exp(x)*(1/cos(x) + tan(x))^n))[n+1] \\ Iain Fox, Dec 20 2017

Formula

a(n) ~ c * d^n * n^n, where d = 1.12712316036287986633533456353714856005183790513784733... and c = 1.61865092826915643845148401952113086265743345... - Vaclav Kotesovec, Dec 21 2017

A298244 a(n) = n! * [x^n] (sec(x) + tan(x))^n.

Original entry on oeis.org

1, 1, 4, 30, 320, 4400, 74016, 1472240, 33800192, 879646464, 25589632000, 822867814912, 28982612385792, 1109640105852928, 45884917741981696, 2038015221365667840, 96765606087737999360, 4890992991269273403392, 262201755871367895711744, 14859866059895961631981568, 887693530413229611155456000
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 15 2018

Keywords

Links

Crossrefs

Cf. A000111, A001250, A217066, A292758, A292976, A296793.
Main diagonal of A322267.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[(Sec[x] + Tan[x])^n, {x, 0, n}], {n, 0, 20}]

Formula

a(n) ~ n! * c * d^n / sqrt(n), where d = 3.063838405249746742835218125232... and c = 0.308377491585051819518915209... - Vaclav Kotesovec, Aug 17 2019

A376878 Triangle read by rows: T(n, k) = n^k * n! * [x^k][y^n]((sec(y) + tan(y)) * exp(x*y)).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 2, 9, 27, 27, 5, 32, 96, 256, 256, 16, 125, 500, 1250, 3125, 3125, 61, 576, 2700, 8640, 19440, 46656, 46656, 272, 2989, 16464, 60025, 168070, 352947, 823543, 823543, 1385, 17408, 109312, 458752, 1433600, 3670016, 7340032, 16777216, 16777216
Offset: 0

Views

Author

Peter Luschny, Oct 13 2024

Keywords

Examples

			Triangle starts:
  [0]    1;
  [1]    1,     1;
  [2]    1,     4,      4;
  [3]    2,     9,     27,     27;
  [4]    5,    32,     96,    256,     256;
  [5]   16,   125,    500,   1250,    3125,    3125;
  [6]   61,   576,   2700,   8640,   19440,   46656,   46656;
  [7]  272,  2989,  16464,  60025,  168070,  352947,  823543,   823543;
  [8] 1385, 17408, 109312, 458752, 1433600, 3670016, 7340032, 16777216, 16777216;

Crossrefs

Cf. A000111, A000312, A079901, A109449, A292976 (row sums).

Programs

  • Maple
    P := n -> coeff(series((sec(y) + tan(y)) * exp(x*y), y, 12), y, n):
seq(seq(coeff(P(n), x,  k) * n^k * n!, k = 0..n), n = 0..8);
T := (n, k) -> ifelse(n = k, n^n, (-1)^binomial(n - k, 2)*n^k*binomial(n, k)*(euler(n - k) - euler(n - k, 0)*2^(n - k))):
seq(print([n], seq(T(n, k), k = 0..n)), n = 0..8);
  • Python
    from math import comb, isqrt
from sympy import bernoulli, euler
def A000111(n): return abs(((1<A376878(n): return comb(a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),b:=n-comb(a+1,2))*a**b*A000111(a-b) # Chai Wah Wu, Nov 13 2024

Formula

T(n, k) = (-1)^binomial(n-k, 2)*n^k*binomial(n, k)*(Euler(n-k) - Euler(n-k, 0)*2^(n - k)) for 0 <= k < n and n^n for n = k.
T(n, k) = n^k*A109449(n, k) = n^k*binomial(n, k)*A000111(n - k).
Showing 1-4 of 4 results.

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