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: Mélika Tebni

Mélika Tebni's wiki page.

Mélika Tebni has authored 27 sequences. Here are the ten most recent ones:

A385641 Partial sums of A097893.

Original entry on oeis.org

1, 3, 8, 20, 51, 133, 356, 972, 2695, 7557, 21372, 60840, 174097, 500295, 1442720, 4172752, 12099411, 35161001, 102375400, 298586652, 872177273, 2551118623, 7471195500, 21904500500, 64286141881, 188844619563, 555216323396, 1633658183432, 4810340397375, 14173698242137
Offset: 0

Views

Author

Mélika Tebni, Aug 03 2025

Keywords

Comments

Second partial sums of the central trinomial coefficients (A002426).
Third partial sums of A025178 (sequence starting 1, 0, 2, 4, 12, 32, 90 .... with offset 0).
For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).
For p Pythagorean prime (A002144), a(p) - 3 == 0 (mod p).
Sequences with g.f. (1-x)^k / sqrt(1-2*x-3*x^2): this sequence (k=-2), A097893 (k=-1), A002426 (k=0), A025178 (k=1), A024997 (k=2), A026083 (k=3). - Mélika Tebni, Aug 25 2025

Crossrefs

Cf. A002144, A002145, A002426, A024997, A025178, A026083, A097893, A128014, A247287, A383527.

Programs

  • Maple
    a := series(exp(x)*(BesselI(0, 2*x) + 2*int(BesselI(0, 2*x), x) + int(int(BesselI(0, 2*x), x), x)), x = 0, 30): seq(n!*coeff(a, x, n), n = 0 .. 29);
  • PARI
    a(n) = sum(k=0, n, sum(i=0, k, sum(j=0, i, binomial(i, i-j)*binomial(j, i-j)))); \\ Michel Marcus, Aug 06 2025
  • Python
    from math import comb as C
def a(n):
    return sum(C(n+1, k+1)*C(2*(k//2), k//2) for k in range(n + 1))
print([a(n) for n in range(30)])

Formula

G.f.: (1 / sqrt((1 + x)*(1 - 3*x))) / (1 - x)^2.
E.g.f.: exp(x)*(BesselI(0, 2*x) + 2*g(x) + Integral_{x=-oo..oo} g(x) dx) where g(x) = Integral_{x=-oo..oo} BesselI(0, 2*x) dx.
D-finite with recurrence n*a(n) = (4*n-1)*a(n-1) - (2*n+1)*a(n-2) - (4*n-5)*a(n-3) + 3*(n-1)*a(n-4).
a(0) = 1, a(1) = 3 and a(n) = a(n-2) - 1 + 2*A383527(n) for n >= 2.
a(n) = Sum_{k=0..n} binomial(n+1, k+1)*A128014(k).
a(n) = Sum_{k=0..n} (2*A247287(k) + k+1).
a(n) ~ 3^(n + 5/2) / (8*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 03 2025

A383609 Triangle read by rows: T(n,k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k) for 0 < k < n, T(n,0) = T(n,n) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 8, 1, 1, 5, 13, 20, 17, 1, 1, 6, 19, 38, 50, 38, 1, 1, 7, 26, 63, 107, 126, 89, 1, 1, 8, 34, 96, 196, 296, 322, 216, 1, 1, 9, 43, 138, 326, 588, 814, 834, 539, 1, 1, 10, 53, 190, 507, 1052, 1728, 2236, 2187, 1374, 1
Offset: 0

Views

Author

Mélika Tebni, May 02 2025

Keywords

Examples

			Triangle T(n, k) starts:
n\k :     0       1       2        3        4       5       6       7
 ====================================================================
  0 :     1
  1 :     1       1
  2 :     1       2       1
  3 :     1       3       4        1
  4 :     1       4       8        8        1
  5 :     1       5      13       20       17       1
  6 :     1       6      19       38       50      38       1
  7 :     1       7      26       63      107     126      89      1
  ...

Crossrefs

Cf. A038185, A167630, A211278.

Programs

  • Maple
    T := proc (n, k) option remember; if k = n or k = 0 then 1 elif k < 0 then 0 else T(n-1, k-2)+T(n-1, k-1)+T(n-1, k) end if end proc:
seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 8);

Formula

Sum_{k=0..n} 2^(n-k)*(T(n, k)(mod 2)) = A038185(n).
Sum_{j=0..n}(Sum_{k=0..j} T(j, k)) = A211278(n).
T(n,k) = A167630(n,n-k).

A383527 Partial sums of A005773.

Original entry on oeis.org

1, 2, 4, 9, 22, 57, 153, 420, 1170, 3293, 9339, 26642, 76363, 219728, 634312, 1836229, 5328346, 15494125, 45137995, 131712826, 384900937, 1126265986, 3299509114, 9676690939, 28407473191, 83470059532, 245465090758, 722406781935, 2127562036990, 6270020029353
Offset: 0

Views

Author

Mélika Tebni, Apr 29 2025

Keywords

Comments

For p prime of the form 4*k+3 (A002145), a(p) == 0 (mod p).
For p Pythagorean prime (A002144), a(p) - 2 == 0 (mod p).
a(n) (mod 2) = A010059(n).
a(A000069(n+1)) is even.
a(A001969(n+1)) is odd.

Links

Crossrefs

Cf. A000069, A001969, A002144, A002145, A005773, A005774, A005775, A010059, A097893, A122896, A167630, A210736, A211278, A257520.

Programs

  • Maple
    gf := (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)):
a := n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n = 0 .. 29);
# Recurrence:
a:= proc(n) option remember; `if`(n<=2, 2^n, 3*a(n-1) - (6/n-1)*a(n-2) + (6/n-3)*a(n-3)) end:
seq(a(n), n = 0 .. 29);
  • Mathematica
    Module[{a, n}, RecurrenceTable[{a[n] == 3*a[n-1] - (6-n)*a[n-2]/n + 3*(2-n)*a[n-3]/n, a[0] == 1, a[1] == 2, a[2] == 4}, a, {n, 0, 30}]] (* Paolo Xausa, May 05 2025 *)
  • Python
    from math import comb as C
def a(n):
  return sum(C(n, k)*abs(sum((-1)**j*C(k, j) for j in range(k//2 + 1))) for k in range(n + 1))
print([a(n) for n in range(30)])

Formula

First differences of A211278.
a(n) = Sum_{k=0..n} A167630(n, k).
Binomial transform of A210736 (see Python program).
G.f.: (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)).
E.g.f.: (Integral_{x=-oo..oo} BesselI(0,2*x) dx + (1 + BesselI(0,2*x)) / 2)*exp(x).
Recurrence: n*a(n) = 3*n*a(n-1) - (6-n)*a(n-2) + 3*(2-n)*a(n-3). If n <= 2, a(n) = 2^n.
a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, May 02 2025
From Mélika Tebni, May 09 2025: (Start)
a(n) = A257520(n) + A097893(n-1) for n > 0.
a(n) = Sum_{j=0..n}(Sum_{k=0..j} A122896(j, k)).
a(n+2) - 3*a(n+1) + 2*a(n) = A005774(n).
a(n+2) - 4*a(n+1) + 4*a(n) - a(n-1) = A005775(n) for n >= 3. (End)

A381889 Expansion of e.g.f.: (BesselI(0, 2*x) + BesselI(1, 2*x))^2*exp(2*x).

Original entry on oeis.org

1, 4, 18, 86, 428, 2192, 11468, 60986, 328532, 1788368, 9819128, 54302712, 302157424, 1690193728, 9497996152, 53588976802, 303434431108, 1723578967056, 9818195961512, 56071829010968, 320970950634288, 1841213871449152, 10582333064327824, 60929582362628968, 351385363433883472
Offset: 0

Views

Author

Mélika Tebni, Mar 09 2025

Keywords

Comments

Binomial transform of A151093.
For p prime, a(p) - 2 == 0 (mod 2*p).

Crossrefs

Cf. A001405, A001700, A005566, A151093.

Programs

  • Maple
    a := n-> add(binomial(n, k)*binomial(n-k, iquo(n-k,2))*binomial(2*k+1,k+1), k = 0 .. n): seq(a(n), n = 0 .. 24);
  • Mathematica
    len := 24; Table[n!,{n, 0, len}] CoefficientList[Series[(BesselI[0, 2x] + BesselI[1, 2x])^2 Exp[2x], {x, 0, len}], x]  (* Peter Luschny, Mar 19 2025 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace((besseli(0, 2*x) + x*besseli(1, 2*x))^2*exp(2*x))) \\ Michel Marcus, Mar 11 2025
  • Python
    from math import comb as C
def a(n):
    return sum(C(n, k)*2**(n-k)*C(k, k//2)*C(k+1, (k+1)//2) for k in range(n+1))
print([a(n) for n in range(25)])

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*A005566(k).
a(n) = Sum_{k=0..n} binomial(n, k)*A001405(n-k)*A001700(k).
a(n) = Sum_{k=0..n} binomial(n, k)*A005773(n-k+1)*A005773(k+1). - Mélika Tebni, Mar 19 2025

A372611 Expansion of (1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x)).

Original entry on oeis.org

1, 7, 26, 90, 310, 1082, 3844, 13892, 50950, 189130, 708876, 2677452, 10175356, 38863780, 149045960, 573559240, 2213551430, 8563950250, 33203854460, 128978378620, 501839077460, 1955475615820, 7629823818680, 29805375256120, 116558646378140, 456270710243332
Offset: 0

Views

Author

Mélika Tebni, May 07 2024

Keywords

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 7 == 0 (mod p).
Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 3 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028322, A029759, A082590, A126966, A188622, A372239, A372420.

Programs

  • Maple
    a := n -> -2^(n-1)*5*I + binomial(2*n, n)*(1-5/2*hypergeom([1, n+1/2], [n+1], 2)): seq(simplify(a(n)), n = 0 .. 25);
  • PARI
    my(x='x+O('x^30)); Vec((1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ Michel Marcus, May 07 2024

Formula

a(n) = 6*A000984(n) - 5* A029759(n) = binomial(2*n,n) + 5*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
a(n) = 2*a(n-1) + A028322(n) = 2*a(n-1) + binomial(2*n, n) + 3*binomial(2*n-2, n-1) for n >= 1.
a(n) = - 2^(n-1)*5*i + binomial(2*n,n)*(1-5/2*hypergeom([1, n + 1/2], [n + 1], 2)).
a(n) = 3*A082590(n-1) + A082590(n) for n >= 1.
a(n) = (7*A188622(n) - 4*A126966(n))/3.
a(n) = 2*A372239(n) - A372420(n).

A372420 Expansion of (1 + x) / ((1 - 2*x)*sqrt(1 - 4*x)).

Original entry on oeis.org

1, 5, 18, 62, 214, 750, 2676, 9708, 35718, 132926, 499228, 1888644, 7186876, 27478508, 105474216, 406182552, 1568563014, 6071812638, 23552366796, 91525132692, 356242058004, 1388588519268, 5419533876696, 21176597444712, 82834229300124, 324326668721100
Offset: 0

Views

Author

Mélika Tebni, Apr 30 2024

Keywords

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 5 == 0 (mod p).
Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028270, A029759, A082590, A126966, A188622.

Programs

  • Maple
    a := n -> -2^(n-1)*3*I + binomial(2*n, n)*(1-3/2*hypergeom([1, n+1/2], [n+1], 2)):
seq(simplify(a(n)), n = 0 .. 25);
  • PARI
    my(x='x+O('x^40)); Vec((1 + x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ Michel Marcus, Apr 30 2024

Formula

a(n) = 4*A000984(n) - 3* A029759(n) = binomial(2*n,n) + 3*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
a(n) = 2*a(n-1) + A028270(n) = 2*a(n-1) + binomial(2*n, n) + binomial(2*n-2, n-1) for n >= 1.
a(n) = - 2^(n-1)*3*i + binomial(2*n,n)*(1-3/2*hypergeom([1,n+1/2],[n + 1],2)).
a(n) = A082590(n-1) + A082590(n) for n >= 1.
a(n) = (5*A188622(n) - 2*A126966(n)) / 3.
D-finite with recurrence n*a(n) -5*n*a(n-1) +2*(n+5)*a(n-2) +4*(2*n-5)*a(n-3)=0. - R. J. Mathar, May 01 2024

A372239 Expansion of (1 + 2*x) / ((1 - 2*x)*sqrt(1 - 4*x)).

Original entry on oeis.org

1, 6, 22, 76, 262, 916, 3260, 11800, 43334, 161028, 604052, 2283048, 8681116, 33171144, 127260088, 489870896, 1891057222, 7317881444, 28378110628, 110251755656, 429040567732, 1672032067544, 6524678847688, 25490986350416, 99696437839132, 390298689482216
Offset: 0

Views

Author

Mélika Tebni, Apr 23 2024

Keywords

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 6 == 0 (mod p).
Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 2 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028283, A029759, A082590, A126966, A188622.

Programs

  • Maple
    a := n -> binomial(2*n,n) + 4*add(2^(n-k-1)*binomial(2*k,k), k = 0 .. n-1):
seq(a(n), n = 0 .. 25);
# Second program:
a:= proc(n) option remember; `if`(n=0,1,2*a(n-1)+2*binomial(2*n-2, n-1)*(3*n-1)/n) end: seq(a(n), n = 0 .. 25);
# Recurrence:
a := proc(n) option remember; if n < 2 then return [1, 6][n + 1] fi;
((-18*(n - 2)^2 - 42*n + 66)*a(n - 1) + 4*(3*n - 1)*(2*n - 3)*a(n - 2)) / (n*(4 - 3*n)) end: seq(a(n), n = 0..25);  # Peter Luschny, Apr 23 2024

Formula

a(n) = 5*A000984(n) - 4* A029759(n) = binomial(2*n,n) + 4*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
a(n) = 2*a(n-1) + A028283(n) = 2*a(n-1) + 2*binomial(2n-2, n-1)*(3*n-1)/n for n >= 1.
a(n) = 2*A082590(n-1) + A082590(n) for n >= 1.
a(n) = 2*A188622(n) - A126966(n).
D-finite with recurrence n*a(n) +2*(-2*n-1)*a(n-1) +4*(-n+6)*a(n-2) +8*(2*n-5)*a(n-3)=0. - R. J. Mathar, Apr 24 2024
E.g.f.: exp(2*x)*(BesselI(0, 2*x)*(1 + 4*x + 2*Pi*x*StruveL(1, 2*x)) - 2*Pi*x*BesselI(1, 2*x)*StruveL(0, 2*x)). - Stefano Spezia, Aug 29 2025

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)|.

A364822 Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)).

Original entry on oeis.org

1, 2, 9, 56, 465, 4832, 60249, 876416, 14570145, 272502272, 5662834089, 129446475776, 3228012339825, 87205172928512, 2537079010567929, 79084060649947136, 2629496833837277505, 92893490657046167552, 3474733464040954877769, 137195165161622584426496, 5702069567580948171751185
Offset: 0

Views

Author

Mélika Tebni, Nov 07 2023

Keywords

Comments

Conjectures: For p prime (p > 2), a(p) == 2 (mod p).
For n = 2^m (m natural number), a(n) == 1 (mod n).

Links

Crossrefs

Cf. A000556, A000557, A005923, A332257, A341724.

Programs

  • Maple
    a := n -> add(binomial(n,2*k)*add(j!*combinat[fibonacci](j+2)*Stirling2(n-2*k,j), j=0..n-2*k), k=0..floor(n/2)):
seq(a(n), n = 0 .. 20);
# second program:
b := proc(n) option remember; `if`(n = 0, 1, 2+2*add(binomial(n,2*k-1)*b(n-2*k+1), k=1..floor((n+1)/2))) end:
a := proc(n) `if`(n = 0, 1, b(n)/2) end: seq(a(n), n = 0 .. 20);
# third program:
(1/2)*((exp(-x) + exp(x))/(1 + exp(-x) - exp(x))): series(%, x, 21):
seq(n!*coeff(%, x, n), n = 0..20);  # Peter Luschny, Nov 07 2023
  • Mathematica
    a[n_]:=n!*SeriesCoefficient[Cosh[x]/(1 - 2*Sinh[x]),{x,0,n}]; Array[a,21,0] (* Stefano Spezia, Nov 07 2023 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(cosh(x) / (1 - 2*sinh(x)))) \\ Michel Marcus, Nov 07 2023

Formula

a(n) = A000556(n) + A332257(n), for n > 0.
a(n) = (-1)^n*Sum_{k=0..floor(n/2)} A341724(n,2*k).
a(n) = (A000556(n) + A005923(n)) / 2.
a(n) ~ n! / (2*log((1 + sqrt(5))/2)^(n+1)).

A366133 Triangle read by rows: coefficients in expansion of another Asveld's polynomials Pi_j(x).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 8, 9, 3, 1, 50, 32, 18, 4, 1, 214, 250, 80, 30, 5, 1, 2086, 1284, 750, 160, 45, 6, 1, 11976, 14602, 4494, 1750, 280, 63, 7, 1, 162816, 95808, 58408, 11984, 3500, 448, 84, 8, 1, 1143576, 1465344, 431136, 175224, 26964, 6300, 672, 108, 9, 1, 20472504, 11435760, 7326720, 1437120, 438060, 53928, 10500, 960, 135, 10, 1
Offset: 0

Views

Author

Mélika Tebni, Sep 30 2023

Keywords

Comments

First negative term is T(35,0) = -230450728485788167742674544892530875760640.
Conjectures: 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).

Examples

			Triangle begins:
      1,
      1,     1,
      3,     2,    1,
      8,     9,    3,    1,
     50,    32,   18,    4,   1,
    214,   250,   80,   30,   5,  1,
   2086,  1284,  750,  160,  45,  6,  1,
  11976, 14602, 4494, 1750, 280, 63,  7,  1,
  ...

Links

Crossrefs

Cf. A000045, A005444 (col 0), A005445, A039948, A048994, A305923 (row sums).

Programs

  • Maple
    T := (n, k) -> binomial(n,k)*add(j!*combinat[fibonacci](j+1)*Stirling1(n-k,j), j=0 .. n-k): seq(print(seq(T(n, k), k = 0 .. n)), n=0 .. 9);
# second Maple program:
T := (n, k) -> add(Stirling2(j, k)/j!*add(i!*combinat[fibonacci](i-j+1)*Stirling1(n, i), i = j .. n), j = k .. n): seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);

Formula

T(n,k) = binomial(n,k)*A005444(n-k).
Sum_{k=1..n} (-1)^(k-1)*(k-1)!*T(n, k) = A005445(n).
E.g.f. of column k: x^k / ((1-log(1+x)-log(1+x)^2)*k!), k >= 0.
Recurrence: T(n,0) = A005444(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)*A039948(i,j)).

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