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.

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)

A341727 Column 1 of A341724.

Original entry on oeis.org

1, -4, 24, -200, 2080, -25932, 377216, -6271120, 117287424, -2437334420, 55714920448, -1389365372760, 37533886128128, -1091981490075868, 34038486791454720, -1131758947520249120, 39982188936149204992, -1495556350390047594276, 59050025742595595436032
Offset: 1

Views

Author

N. J. A. Sloane, Mar 04 2021

Keywords

Crossrefs

Cf. A000556, A000557, A341724 ,A341726, A341728.

Programs

  • Maple
    A341727 := n -> (-1)^(n-1)*n*add(k!*combinat[fibonacci](k+2)*Stirling2(n-1, k), k=0..n-1):seq(A341727(n), n = 1 .. 19); # Mélika Tebni, Sep 04 2023
# E.g.f. Maple program:
A341727 := series(x / (1 + 2*sinh(x)), x = 0, 20):
seq(n!*coeff(A341727, x, n), n = 1 .. 19); # Mélika Tebni, Sep 04 2023

Formula

From Mélika Tebni, Sep 04 2023: (Start)
E.g.f.: x / (1 + 2*sinh(x)).
a(n) = (-1)^(n-1)*n*A000557(n-1).
a(n) = (-1)^(n-1)*Sum_{k=0..n} A000556(k)*(n-k)*binomial(n, k). (End)

Extensions

More terms from Mélika Tebni, Sep 04 2023

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
Showing 1-4 of 4 results.

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

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