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 1-5 of 5 results.

A341728 Column 1 of A341725.

Original entry on oeis.org

1, 6, 39, 324, 3365, 41958, 610351, 10146888, 189775017, 3943689930, 90148635203, 2248040395692, 60731103481789, 1766863166037102, 55075428554246295, 1831224444159278736, 64692540643308320081, 2419861007021854813074, 95544948688075940395627
Offset: 1

Views

Author

N. J. A. Sloane, Mar 04 2021

Keywords

Crossrefs

Cf. A000557, A005923, A341725, A341726, A341727.

Programs

  • Maple
    A341728 := n -> add((n-k)*binomial(n, k)add(j!*combinat[fibonacci](j+2)*Stirling2(k,j), j=0..k), k=0..n):seq(A341728(n), n=1.. 19); # Mélika Tebni, Sep 04 2023
# E.g.f. Maple program:
A341728 := series(x*exp(x) / (1 - 2*sinh(x)), x = 0, 20):
seq(n!*coeff(A341728, x, n), n = 1 .. 19); # Mélika Tebni, Sep 04 2023

Formula

From Mélika Tebni, Sep 04 2023: (Start)
a(n) = n*A005923(n-1).
E.g.f.: x*exp(x) / (1 - 2*sinh(x)).
a(n) = Sum_{k=0..n} (n-k)*binomial(n, k)*A000557(k). (End)

Extensions

More terms from Mélika Tebni, Sep 04 2023

A341724 Triangle read by rows: coefficients of expansion of certain sums P_2(n,k) of Fibonacci numbers as a sum of powers.

Original entry on oeis.org

1, -2, 1, 8, -4, 1, -50, 24, -6, 1, 416, -200, 48, -8, 1, -4322, 2080, -500, 80, -10, 1, 53888, -25932, 6240, -1000, 120, -12, 1, -783890, 377216, -90762, 14560, -1750, 168, -14, 1, 13031936, -6271120, 1508864, -242032, 29120, -2800, 224, -16, 1
Offset: 0

Views

Author

N. J. A. Sloane, Mar 04 2021

Keywords

Comments

Conjectures from Mélika Tebni, Sep 04 2023: (Start)
For 0 < k < p and p prime, T(p,k) == 0 (mod p).
For 0 <= k < n and n = 2^m (m natural number), T(n,k) == 0 (mod n). (End)

Examples

			Triangle begins:
         1;
        -2,        1;
         8,       -4,       1;
       -50,       24,      -6,       1;
       416,     -200,      48,      -8,     1;
     -4322,     2080,    -500,      80,   -10,     1;
     53888,   -25932,    6240,   -1000,   120,   -12,   1;
   -783890,   377216,  -90762,   14560, -1750,   168, -14,   1;
  13031936, -6271120, 1508864, -242032, 29120, -2800, 224, -16, 1;
  ...

References

  • Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin’s summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See Table 3.

Crossrefs

Column 0 is a signed version of A000557, column 1 is A341727.
Cf. A000556, A005923, A341723, A341725, A341726.

Programs

  • Maple
    egf:= k-> x^k / ((1-2*sinh(-x))*k!):
A341724:= (n,k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A341724(n,k), k=0..n)), n=0..8); # Mélika Tebni, Sep 04 2023

Formula

From Mélika Tebni, Sep 04 2023: (Start)
E.g.f. of column k: x^k / ((1-2*sinh(-x))*k!).
T(n,k) = (-1)^(n-k)*binomial(n,k)*A000557(n-k).
Recurrence: T(n,0) = (-1)^n*A000557(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)
From Alois P. Heinz, Sep 04 2023: (Start)
|Sum_{k=0..n} T(n,k)| = A000556(n).
Sum_{k=0..n} |T(n,k)| = A005923(n).
Sum_{k=0..n} k * T(n,k) = A341726(n). (End)

A341723 Triangle read by rows: coefficients of expansion of certain weighted sums P_1(n,k) of Fibonacci numbers as a sum of powers.

Original entry on oeis.org

1, -1, 1, 5, -2, 1, -31, 15, -3, 1, 257, -124, 30, -4, 1, -2671, 1285, -310, 50, -5, 1, 33305, -16026, 3855, -620, 75, -6, 1, -484471, 233135, -56091, 8995, -1085, 105, -7, 1, 8054177, -3875768, 932540, -149576, 17990, -1736, 140, -8, 1
Offset: 0

Views

Author

N. J. A. Sloane, Mar 04 2021

Keywords

Comments

Conjectures from Mélika Tebni, Sep 09 2023: (Start)
For 0 < k < p and p prime, T(p,k) == 0 (mod p).
For 0 < k < n (k odd) and n = 2^m (m natural number), T(n,k) == 0 (mod n). (End)

Examples

			Triangle begins:
        1;
       -1,        1;
        5,       -2,      1;
      -31,       15,     -3,       1;
      257,     -124,     30,      -4,     1;
    -2671,     1285,   -310,      50,    -5,     1;
    33305,   -16026,   3855,    -620,    75,    -6,   1;
  -484471,   233135, -56091,    8995, -1085,   105,  -7,  1;
  8054177, -3875768, 932540, -149576, 17990, -1736, 140, -8, 1;
  ...

References

  • Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin’s summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See Table 2.

Crossrefs

Column 0 is a signed version of A000556, column 1 is A341726.
Cf. A341724, A341725.

Programs

  • Maple
    egf:= k-> exp(x)*x^k / ((1+2*sinh(x))*k!):
A341723:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A341723(n, k), k=0..n)), n=0..8); # Mélika Tebni, Sep 09 2023
second Maple program:
A341723:= (n, k)-> (-1)^(n-k)*binomial(n, k)*add(j!*combinat[fibonacci](j+1)*Stirling2(n-k,j), j=0.. n-k):
seq(print(seq(A341723(n, k), k=0..n)), n=0..8); # Mélika Tebni, Sep 09 2023

Formula

From Mélika Tebni, Sep 09 2023: (Start)
E.g.f. of column k: exp(x)*x^k / ((1+2*sinh(x))*k!).
T(n,k) = (-1)^(n-k)*binomial(n,k)*A000556(n-k).
Recurrence: T(n,0) = (-1)^n*A000556(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1. (End)

A365962 Triangle read by rows: coefficients in expansion of Asveld's polynomials Pi_j(x).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 48, 30, 9, 1, 312, 192, 60, 12, 1, 2520, 1560, 480, 100, 15, 1, 24480, 15120, 4680, 960, 150, 18, 1, 277200, 171360, 52920, 10920, 1680, 210, 21, 1, 3588480, 2217600, 685440, 141120, 21840, 2688, 280, 24, 1, 52254720, 32296320, 9979200, 2056320, 317520, 39312, 4032, 360, 27, 1
Offset: 0

Views

Author

Mélika Tebni, Sep 23 2023

Keywords

Examples

			Triangle begins:
      1,
      3,     1,
     10,     6,    1,
     48,    30,    9,   1,
    312,   192,   60,  12,   1,
   2520,  1560,  480, 100,  15,  1,
  24480, 15120, 4680, 960, 150, 18, 1,
  ...

Links

Crossrefs

Cf. A000045, A005921 (col 0), A005922 (col 1), A039948, A341725.

Programs

  • Maple
    T := proc(n, k) option remember;if k = n then 1  else (n!/k!*combinat[fibonacci](n-k+3)) fi end: seq(print(seq(T(n, k), k = 0..n)), n=0..9);
# second Maple program:
T := (n,k) -> add(Stirling2(j, k)*add(Stirling1(n, i)*A341725(i, j), i = j .. n), j = k .. n): seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);
  • PARI
    T(n,k) = n!/k!*sum(j=k, n, fibonacci(j-k+1)*binomial(2,n-j)) \\ Winston de Greef, Oct 21 2023
  • PARI
    T(n,k) = if(n == k, 1, n!/k!*fibonacci(n-k+3)) \\ Winston de Greef, Oct 21 2023

Formula

E.g.f. of column k: (1+x)^2*x^k / ((1-x-x^2)*k!), k >= 0.
T(n,n) = 1 and T(n,k) = n!/k!*Fibonacci(n-k+3), n > k >= 0.
T(n,k) = n!/k!*Sum_{j=k..n} Fibonacci(j-k+1)*binomial(2,n-j).
T(n,k) = n!/k!*Sum_{j=k..n} (Fibonacci(j-k)+(-1)^(j-k))*binomial(3,n-j).
Recurrence: T(n,0) = A005921(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1.
T(n,k) = Sum_{j=k..n} Stirling2(j,k)*(Sum_{i=j..n} Stirling1(n,i)*A341725(i,j)).
Sum_{j=k..n} (-1)^(n-j)*(n-j+1)!*binomial(n,j)*T(j,k) = A039948(n,k).

A367887 Expansion of e.g.f. exp(2*x) / (1 - 2*sinh(x)).

Original entry on oeis.org

1, 4, 20, 130, 1088, 11314, 141080, 2052250, 34118048, 638102434, 13260323240, 303117147370, 7558845354608, 204203189722354, 5940927689713400, 185186461979970490, 6157337034085736768, 217523186522883467074, 8136577601614291359560, 321261794453042025993610, 13352198666907246870560528
Offset: 0

Views

Author

Mélika Tebni, Dec 04 2023

Keywords

Links

Crossrefs

Cf. A000045, A000557, A005923, A341723, A341724, A341725.

Programs

  • Maple
    a := n -> -1-0^n+add(k!*combinat[fibonacci](k+4)*Stirling2(n, k), k = 0 .. n):
seq(a(n), n=0..20);
# second program:
a := proc(n) option remember; `if`(n=0,1,3^n+add((2^(n-k)-1)*binomial(n, k)*a(k), k=0..n-1)) end:
seq(a(n), n=0..20);
# third program:
a := n -> add(2^k*binomial(n, k)*add(j!*combinat[fibonacci](j+2)*Stirling2(n-k, j), j=0..n-k), k=0..n):
seq(a(n), n=0..20);
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(2*x) / (1 - 2*sinh(x)))) \\ Michel Marcus, Dec 04 2023

Formula

a(n) = Sum_{k=0..n} A341725(n,k).
a(n) = (-1)^n*Sum_{k=0..n} (-2)^k*A341724(n,k).
a(n) = -1-0^n+Sum_{k=0..n} k!*Fibonacci(k+4)*Stirling2(n,k).
a(0) = 1; a(n) = 3^n+Sum_{k=0..n-1} (2^(n-k)-1)*binomial(n,k)*a(k).
a(n) ~ n! * (phi)^2 / (sqrt(5) * (log(phi))^(n+1)), where phi is the golden ratio.
a(n) = -1 + A000557(n) + A005923(n) = -1 + Sum_{k=0..n} |A341723(n,k) + A341724(n,k)|.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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