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 1-6 of 6 results.

A144387 Triangle read by rows: row n gives the coefficients in the expansion of Sum_{j=0..n} A000040(j+1)*x^j*(1 - x)^(n - j).

Original entry on oeis.org

2, 2, 1, 2, -1, 4, 2, -3, 5, 3, 2, -5, 8, -2, 8, 2, -7, 13, -10, 10, 5, 2, -9, 20, -23, 20, -5, 12, 2, -11, 29, -43, 43, -25, 17, 7, 2, -13, 40, -72, 86, -68, 42, -10, 16, 2, -15, 53, -112, 158, -154, 110, -52, 26, 13, 2, -17, 68, -165, 270, -312, 264, -162, 78, -13, 18
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

Keywords

Comments

Row sums yield the primes A000040.

Examples

			Triangle begins
    2;
    2,   1;
    2,  -1,  4;
    2,  -3,  5,    3;
    2,  -5,  8,   -2,   8;
    2,  -7, 13,  -10,  10,    5;
    2,  -9, 20,  -23,  20,   -5,  12;
    2, -11, 29,  -43,  43,  -25,  17,    7;
    2, -13, 40,  -72,  86,  -68,  42,  -10, 16;
    2, -15, 53, -112, 158, -154, 110,  -52, 26,  13;
    2, -17, 68, -165, 270, -312, 264, -162, 78, -13, 18;
    ...

Links

Crossrefs

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144400, A174128.

Programs

  • Mathematica
    p[x_, n_] = Sum[Prime[k + 1]*x^k*(1 - x)^(n - k), {k, 0, n}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
  • Sage
    def p(n,x): return sum( nth_prime(j+1)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (0..12)] # G. C. Greubel, Jul 15 2021

Extensions

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 19 2018

A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)*binomial(n, j)*x^(j - 1)*(1 - x)^(n - j).

Original entry on oeis.org

1, 2, -1, 3, -3, 1, 4, -6, 4, 0, 5, -10, 10, 0, -3, 6, -15, 20, 0, -18, 10, 7, -21, 35, 0, -63, 70, -24, 8, -28, 56, 0, -168, 280, -192, 49, 9, -36, 84, 0, -378, 840, -864, 441, -89, 10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145, 11, -55, 165, 0
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 03 2008

Keywords

Examples

			Triangle begins:
    1;
    2,  -1;
    3,  -3,   1;
    4,  -6,   4, 0;
    5, -10,  10, 0,   -3;
    6, -15,  20, 0,  -18,   10;
    7, -21,  35, 0,  -63,   70,   -24;
    8, -28,  56, 0, -168,  280,  -192,   49;
    9, -36,  84, 0, -378,  840,  -864,  441,  -89;
   10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145;
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 22 2018

Links

Crossrefs

Row sums: subsequence of A000931, A078027, A182097.
Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A174128.

Programs

  • Mathematica
    a[n_]:= a[n]= If[n<3, Fibonacci[n], a[n-2] + a[n-3]];
p[x_, n_]:= Sum[a[k]*Binomial[n, k]*x^(k-1)*(1-x)^(n-k), {k, 0, n}];
Table[Coefficient[p[x, n], x, k], {n, 12}, {k, 0, n-1}]//Flatten
  • Sage
    @CachedFunction
def f(n): return fibonacci(n) if (n<3) else f(n-2) + f(n-3)
def p(n,x): return sum( binomial(n,j)*f(j)*x^(j-1)*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (1..12)] # G. C. Greubel, Jul 14 2021

Formula

G.f.: (y - (1 - 2*x)*y^2)/(1 - 3*(1 - x)*y + (3 - 6*x + 2*x^2)*y^2 - (1 - 3*x + 2*x^2 + x^3)*y^3). - Franck Maminirina Ramaharo, Oct 22 2018

Extensions

Edited, and new name by Franck Maminirina Ramaharo, Oct 22 2018

A174128 Irregular triangle read by rows: row n (n > 0) is the expansion of Sum_{m=1..n} A001263(n,m)*x^(m - 1)*(1 - x)^(n - m).

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 3, -3, 1, 6, -4, -4, 2, 1, 10, 0, -20, 10, 1, 15, 15, -55, 15, 15, -5, 1, 21, 49, -105, -35, 105, -35, 1, 28, 112, -140, -266, 364, -56, -56, 14, 1, 36, 216, -84, -882, 756, 336, -504, 126, 1, 45, 375, 210, -2100, 672, 2520, -2100, 210, 210, -42
Offset: 1

Views

Author

Roger L. Bagula, Mar 09 2010

Keywords

Comments

Row n gives the coefficients in the expansion of (1/x)*(1 - x)^n*N(n,x/(1 - x)), where N(n,x) is the n-th row polynomial for the triangle of Narayana numbers A001263.

Examples

			Triangle begins
    1;
    1;
    1,  1,  -1;
    1,  3,  -3;
    1,  6,  -4,   -4,    2;
    1, 10,   0,  -20,   10;
    1, 15,  15,  -55,   15,  15,  -5;
    1, 21,  49, -105,  -35, 105, -35;
    1, 28, 112, -140, -266, 364, -56,  -56,  14;
    1, 36, 216,  -84, -882, 756, 336, -504, 126;
    ...

Links

Crossrefs

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A144400.

Programs

  • Mathematica
    p[x_, n_]:= p[x, n]= Sum[(Binomial[n, j]*Binomial[n, j-1]/n)*x^j*(1-x)^(n-j), {j, 1, n}]/x;
Table[CoefficientList[p[x, n], x], {n, 1, 12}]//Flatten
  • Sage
    def p(n,x): return (1/(n*x))*sum( binomial(n,j)*binomial(n,j-1)*x^j*(1-x)^(n-j) for j in (1..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (1..12)] # G. C. Greubel, Jul 14 2021

Formula

The n-th row of the triangle is generated by the coefficients of (1 - x)^(n - 1)*F(-n, 1 - n; 2; x/(1 - x)), where F(a, b ; c; z) is the ordinary hypergeometric function.
G.f.: (1 - y - sqrt(1 - 2*y + ((1 - 2*x)*y)^2))/(2*(1 - x)*x*y). - Franck Maminirina Ramaharo, Oct 23 2018

Extensions

Edited and new name by Joerg Arndt, Oct 28 2014
Comments and formula clarified by Franck Maminirina Ramaharo, Oct 23 2018

A142073 Irregular triangle, T(n, k) = coefficients of p(x, n), where p(x, n) = (1-2*x)^(n+1) * Sum_{j>=0} j^n*(x/(1-x))^j, read by rows.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, -4, 2, 1, 7, -16, 8, 1, 21, -28, -26, 48, -16, 1, 51, 32, -356, 408, -136, 1, 113, 492, -1774, 1072, 912, -1088, 272, 1, 239, 2592, -5008, -6656, 20736, -15872, 3968, 1, 493, 10628, -50, -94432, 154528, -57856, -45056, 39680, -7936, 1, 1003, 38768, 108820, -621352, 455608, 848384, -1538816, 884480, -176896
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

Keywords

Comments

Except for n=0, the row sums are zero.

Examples

			Irregular triangle begins as:
  1;
  1,  -1;
  1,  -1;
  1,   1,    -4,     2;
  1,   7,   -16,     8;
  1,  21,   -28,   -26,     48,    -16;
  1,  51,    32,  -356,    408,   -136;
  1, 113,   492, -1774,   1072,    912,  -1088,    272;
  1, 239,  2592, -5008,  -6656,  20736, -15872,   3968;
  1, 493, 10628,   -50, -94432, 154528, -57856, -45056, 39680, -7936;

Links

Crossrefs

Cf. A008292, A141720.

Programs

  • Magma
    m:=12;
R:=PowerSeriesRing(Integers(), m+2);
b:= func< n | n eq 0 select 0 else 2*Floor((n+1)/2) -1 >;
Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j)^n: j in [0..k]]) >;
p:= func< n,x | (&+[Eulerian(n,j)*(x-1)^j: j in [0..n]]) >;
T:= func< n,k | Coefficient(R!( p(n,x) ), k) >;
[T(n,n-k): k in [0..b(n)], n in [0..m]]; // G. C. Greubel, May 26 2024
  • Mathematica
    p[x_, n_]= If[n==0, 1, (1-2*x)^(n+1)*Sum[k^n*(x/(1-x))^k, {k,0, Infinity}]/x];
Table[CoefficientList[p[x,n], x], {n,0,12}]//Flatten
  • SageMath
    m=12
def b(n): return 2*int((n+1)/2) - 1 + int(n==0)
def Eulerian(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j)^n for j in range(k+1))
def p(x,n): return sum(Eulerian(n,j)*(x-1)^j for j in range(n+1))
def T(n,k): return ( p(x,n) ).series(x, n+1).list()[k]
flatten([[T(n,n-k) for k in range(b(n)+1)] for n in range(m+1)]) # G. C. Greubel, May 26 2024

Formula

T(n, k) = [x^k]( p(x, n) ), where p(x, n) = (1-2*x)^(n+1) * Sum_{j>=0} j^n*(x/(1-x))^j, or p(x, n) = (1-2*x)^(n+1)*PolyLog(-n, x/(1-x))/x.
T(n, k) = [x^k]( f(x, n) ), where f(x, n) = Sum_{j=0..n} Eulerian(n, j)*(x-1)^j. - Mourad Rahmani (mrahmani(AT)usthb.dz), Aug 29 2010

Extensions

Edited by G. C. Greubel, May 26 2024

A143505 Triangle of coefficients of the polynomials x^(n - 1)*A(n,x + 1/x), where A(n,x) are the Eulerian polynomials of A008292.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 11, 14, 23, 14, 11, 1, 1, 26, 70, 104, 139, 104, 70, 26, 1, 1, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1, 1, 120, 1197, 3016, 5970, 8568, 9549, 8568, 5970, 3016, 1197, 120, 1, 1, 247, 4300, 17101, 37105, 70474, 90069, 107241, 90069
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2008

Keywords

Comments

Row sums yield A000670 (without leading 1).

Examples

			Triangle begins:
   1;
   1,  1,   1;
   1,  4,   3,   4,   1;
   1, 11,  14,  23,  14,  11,   1;
   1, 26,  70, 104, 139, 104,  70,  26,   1;
   1, 57, 307, 530, 973, 947, 973, 530, 307, 57, 1;
    ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018

Links

Crossrefs

Compare with A141720.
Cf. A008292.
Cf. A143506, A143507.

Programs

  • Mathematica
    Table[CoefficientList[FullSimplify[ExpandAll[(1 - x - 1/x)^(n + 1)*x^(n - 1)*PolyLog[-n, x + 1/x]/(x + 1/x)]], x], {n, 1, 10}]//Flatten

Formula

Row n is generated by the polynomial (1 - x - 1/x)^(n + 1)*x^(n - 1)*Li(-n, x + 1/x)/(x + 1/x), where Li(n, z) is the polylogarithm function.
E.g.f.: (exp(x*y) - exp((1 + x^2)*y))/(x*exp((1 + x^2)*y) - (1 + x^2)*exp(x*y)). - Franck Maminirina Ramaharo, Oct 25 2018

Extensions

Edited and new name by Franck Maminirina Ramaharo, Oct 25 2018

A225678 Triangle read by rows, T(n,k) = sum_{j=0..n} (-1)^(n+k+j) A(n,j)*C(j,n-k), A(n,j) the Eulerian numbers; n >= 0, k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 2, -2, 0, 1, 8, -8, 0, 0, 1, 22, -6, -32, 16, 0, 1, 52, 84, -272, 136, 0, 0, 1, 114, 606, -1168, -96, 816, -272, 0, 1, 240, 2832, -2176, -8832, 11904, -3968, 0, 0, 1, 494, 11122, 11072, -83360, 71168, 13312, -31744, 7936, 0, 1, 1004
Offset: 0

Views

Author

Peter Luschny, May 12 2013

Keywords

Examples

			[0]                1
[1]              0, 1
[2]             0, 1, 0
[3]           0, 1, 2, -2
[4]          0, 1, 8, -8, 0
[5]      0, 1, 22, -6, -32, 16
[6]    0, 1, 52, 84, -272, 136, 0

Crossrefs

Cf. A141720.

Programs

  • Maple
    with(combinat): A225678 := (n, k) -> add((-1)^(n+k+j)*eulerian1(n,j) *binomial(j,n-k), j=0..n); seq(print(seq(A225678(n,k),k=0..n)),n=0..8);
Showing 1-6 of 6 results.

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