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.

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A100321 The trinomial transform (A027907) gives powers of 2, while the trinomial transform of this sequence shift one place left gives powers of 3.

Original entry on oeis.org

1, 1, 0, 2, -3, 8, -16, 35, -72, 150, -307, 628, -1276, 2587, -5228, 10546, -21235, 42704, -85784, 172179, -345344, 692286, -1387155, 2778492, -5563748, 11138443, -22294596, 44617850, -89282067, 178639160, -357399712, 714995843, -1430309496, 2861133222, -5723098483, 11447543236
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2004

Keywords

Examples

			2^3 = 1*(1) + 3*(1) + 6*(0) + 7*(2) + 6*(-3) + 3*(8) + 1*(-16).
3^3 = 1*(1) + 3*(0) + 6*(2) + 7*(-3) + 6*(8) + 3*(-16) + 1*(35).

Links

Crossrefs

Cf. A000045, A027907.
Cf. A000079, A000244.

Programs

  • Magma
    [((-1)^n*(3*Fibonacci(n-1) -2^n) +1)/3: n in [0..40]]; // G. C. Greubel, Feb 01 2023
  • Mathematica
    LinearRecurrence[{-2,2,3,-2}, {1,1,0,2}, 41] (* G. C. Greubel, Feb 01 2023 *)
  • PARI
    a(n)=polcoeff((1+3*x-3*x^3)/(1+2*x-2*x^2-3*x^3+2*x^4+x*O(x^n)),n)
  • SageMath
    def A100321(n): return ((-1)^n*(3*fibonacci(n-1) -2^n) +1)/3
[A100321(n) for n in range(41)] # G. C. Greubel, Feb 01 2023

Formula

G.f.: (1 + 3*x - 3*x^3) / (1 + 2*x - 2*x^2 - 3*x^3 + 2*x^4).
2^n = Sum_{k=0..2*n} A027907(n, k)*a(k).
3^n = Sum_{k=0..2*n} A027907(n, k)*a(k+1).
a(n) = (1/3)*((-1)^n*(3*Fibonacci(n-1) - 2^n) + 1). - Ralf Stephan, May 15 2007

A104495 Matrix inverse of triangle A099602, read by rows, where row n of A099602 equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -1, 3, -4, 1, 1, -4, 12, -5, 1, -1, 5, -34, 17, -7, 1, 1, -6, 98, -51, 32, -8, 1, -1, 7, -294, 149, -124, 40, -10, 1, 1, -8, 919, -443, 448, -164, 61, -11, 1, -1, 9, -2974, 1362, -1576, 612, -298, 72, -13, 1, 1, -10, 9891, -4336, 5510, -2188, 1294, -370, 99, -14, 1, -1, 11, -33604, 14227, -19322, 7698
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2005

Keywords

Comments

Row sums are A104496. Absolute row sums form A014137 (partial sums of Catalan numbers). Column 2 is signed A014143.

Examples

			Rows begin:
1;
-1,1;
1,-2,1;
-1,3,-4,1;
1,-4,12,-5,1;
-1,5,-34,17,-7,1;
1,-6,98,-51,32,-8,1;
-1,7,-294,149,-124,40,-10,1;
1,-8,919,-443,448,-164,61,-11,1;
-1,9,-2974,1362,-1576,612,-298,72,-13,1; ...

Crossrefs

Cf. A099602, A027907, A000108, A104496, A014137, A014143.

Programs

  • PARI
    {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));polcoeff(polcoeff( (1+X*Y/(1+X))/(1+X-Y^2*(1-(1+4*X)^(1/2))^2/4),n,x),k,y)}

Formula

G.f.: A(x, y) = (1 + x*y/(1+x))/(1+x - x^2*y^2*Catalan(-x)^2), also G.f.: Column_k(x) = Catalan(-x)^(2*[k/2])/(1+x)^[(k+3)/2], where Catalan(x)=(1-(1-4*x)^(1/2))/(2*x) (cf. A000108).

A136518 a(n) = A027907(2^n, n), where A027907 = triangle of trinomial coefficients.

Original entry on oeis.org

1, 2, 10, 112, 3620, 360096, 116950848, 129755798400, 507413158135840, 7132358041777380352, 364730093112968976177664, 68393665694364347188157159424, 47308574208170527265149009962117120
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2008

Keywords

Comments

This is a special case of the more general statement:
Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =
Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b)
where F(x) = 1+x+x^2, q=2, m=1, b=0.

Examples

			A(x) = 1 + 2*x + 10*x^2 + 112*x^3 + 3620*x^4 + 360096*x^5 + ...
A(x) = 1 + log(1 +2*x +4*x^2) + log(1 +4*x +16*x^2)^2/2! + log(1 +8*x +64*x^2)^3/3! + ...

Links

Crossrefs

Cf. A027907, A136519.

Programs

  • Magma
    m:=40;
gf:= func< x | (&+[Log(1 +2^j*x +4^j*x^2)^j/Factorial(j): j in [0..m+1]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( gf(x) )); // G. C. Greubel, Jul 27 2023
  • Mathematica
    With[{m=40, f= 1 +2^j*x +4^j*x^2}, CoefficientList[Series[ Sum[Log[f]^j/j!, {j,0,m+1}], {x,0,m}], x]] (* G. C. Greubel, Jul 27 2023 *)
  • PARI
    a(n)=polcoeff((1+x+x^2+x*O(x^n))^(2^n),n)
  • PARI
    /* As coefficient x^n of Series: */ a(n)=polcoeff(sum(i=0,n,log(1+2^i*x+2^(2*i)*x^2 +x*O(x^n))^i/i!),n)
  • SageMath
    m=40
def f(x): return sum( log(1 + 2^j*x + 4^j*x^2)^j/factorial(j) for j in range(m+2) )
def A136518_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A136518_list(m) # G. C. Greubel, Jul 27 2023

Formula

a(n) = [x^n] (1 + x + x^2)^(2^n), the coefficient of x^n in (1 + x + x^2)^(2^n).
O.g.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x + 4^n*x^2)^n / n!.

A136519 a(n) = A027907(2^n+1, n), where A027907 = triangle of trinomial coefficients.

Original entry on oeis.org

1, 3, 15, 156, 4556, 417384, 128004240, 136874853504, 523288667468832, 7257782720507161152, 368292386875012729754240, 68761030015590030510485191680, 47447175348985315294381264871833600
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2008

Keywords

Examples

			A(x) = 1 + 3*x + 15*x^2 + 156*x^3 + 4556*x^4 + 417384*x^5 + ...
A(x) = (1 +x +x^2) + (1 +2*x +4*x^2)*log(1 +2*x +4*x^2) + (1 +4*x +16*x^2)*log(1 +4*x +16*x^2)^2/2! + (1 +8*x +64*x^2)*log(1 +8*x +64*x^2)^3/3! + (1 +16*x +256*x^2)*log(1 +16*x +256*x^2)^4/4! + ...
This is a special case of the more general statement:
  Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =
  Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b)
  where F(x) = 1+x+x^2, q=2, m=1, b=1.

Links

Crossrefs

Cf. A027907, A136518.

Programs

  • Magma
    m:=40; // gf of A136519
gf:= func< x | (&+[(1 +2^j*x +4^j*x^2)*Log(1 +2^j*x +4^j*x^2)^j/Factorial(j): j in [0..m+1]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( gf(x) )); // G. C. Greubel, Jul 27 2023
  • Mathematica
    With[{m=40, f= 1 +2^j*x +4^j*x^2}, CoefficientList[Series[ Sum[f*Log[f]^j/j!, {j,0,m+1}], {x,0,m}], x]] (* G. C. Greubel, Jul 27 2023 *)
  • PARI
    a(n)=polcoeff((1+x+x^2+x*O(x^n))^(2^n+1),n)
  • PARI
    /* As coefficient x^n of Series: */ a(n)=polcoeff(sum(i=0,n,(1+2^i*x+2^(2*i)*x^2)*log(1+2^i*x+2^(2*i)*x^2 +x*O(x^n))^i/i!),n)
  • SageMath
    m=40
def f(x): return sum( (1 + 2^j*x + 4^j*x^2)*log(1 + 2^j*x + 4^j*x^2)^j/factorial(j) for j in range(m+2) )
def A136519_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A136519_list(m) # G. C. Greubel, Jul 27 2023

Formula

a(n) = [x^n] (1 + x + x^2)^(2^n+1), the coefficient of x^n in (1 + x + x^2)^(2^n+1).
O.g.f.: A(x) = Sum_{n>=0} (1 + 2^n*x + 4^n*x^2) * log(1 + 2^n*x + 4^n*x^2)^n / n!.

A200475 G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..2*n} A027907(n,k)^2 * x^k * A(x)^(2*k)) * x^n*A(x)^n/n ), where A027907 is the triangle of trinomial coefficients.

Original entry on oeis.org

1, 1, 3, 13, 65, 350, 1981, 11627, 70132, 432090, 2707595, 17202779, 110563543, 717547090, 4695774335, 30952628861, 205318395288, 1369539030021, 9180527051187, 61813112864984, 417850301293691, 2834802846097200, 19294989810689802, 131723105933867817, 901709774424393614
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2011

Keywords

Comments

Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 65*x^4 + 350*x^5 + 1981*x^6 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A^2 + x^2*A^4)*x*A +
(1 + 2^2*x*A^2 + 3^2*x^2*A^4 + 2^2*x^3*A^6 + x^4*A^8)*x^2*A^2/2 +
(1 + 3^2*x*A^2 + 6^2*x^2*A^4 + 7^2*x^3*A^6 + 6^2*x^4*A^8 + 3^2*x^5*A^10 + x^6*A^12)*x^3*A^3/3 +
(1 + 4^2*x*A^2 + 10^2*x^2*A^4 + 16^2*x^3*A^6 + 19^2*x^4*A^8 + 16^2*x^5*A^10 + 10^2*x^6*A^12 + 4^2*x^7*A^14 + x^8*A^16)*x^4*A^4/4 +
(1 + 5^2*x*A^2 + 15^2*x^2*A^4 + 30^2*x^3*A^6 + 45^2*x^4*A^8 + 51^2*x^5*A^10 + 45^2*x^6*A^12 + 30^2*x^7*A^14 + 15^2*x^8*A^16 + 5^2*x^9*A^18 + x^10*A^20)*x^5*A^5/5 +...
which involves the squares of the trinomial coefficients A027907(n,k).

Crossrefs

Cf. A200377, A199248, A186236, A199257, A168592, A027907.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(1-x*A^2+x^3*A^6-x^5*A^10+x^6*A^12)/(1-x*A^2+x*O(x^n))^2);polcoeff(A,n)}
  • PARI
    /* G.f. A(x) using the squares of the trinomial coefficients */
{A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k*(A+x*O(x^n))^(2*k))*x^m*A^m/m))); polcoeff(A, n)}

Formula

G.f. satisfies: A(x) = (1 + x^3*A(x)^6)*(1 + x^6*A(x)^12)/((1 - x*A(x)^2)*(1 - x^4*A(x)^8)).

A027910 T(2n,n-2), T given by A027907.

Original entry on oeis.org

1, 6, 36, 210, 1221, 7098, 41328, 241128, 1409895, 8260934, 48497064, 285219090, 1680166215, 9912297150, 58558256496, 346371955776, 2051126447742, 12158963346852, 72147074769640, 428476010502582, 2546776668682323, 15149061841758174, 90175327717962024
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

a(n) is also the number of lattice paths from (0,0) to (2n-1,n-2) taking north and east steps avoiding north^{>=3}. - Shanzhen Gao, Apr 20 2010

Links

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<3, n*(n-1)/2,
      (14*(2*n-1)*(65*n^3-120*n^2+37*n+6) *a(n-1)
      +36*(n-1)*(2*n-1)*(2*n-3)*(13*n+2) *a(n-2))/
      (3*(13*n-11)*(n-2)*(3*n+2)*(3*n+1)))
    end:
seq(a(n), n=2..25);  # Alois P. Heinz, Aug 07 2013

Formula

a(n) = Sum_{i=0..floor((2*n-3)/2)} C(2*n,n-2-i)*C(n-2-i,i). Shanzhen Gao, Apr 20 2010
G.f.: -g^2*(g^2+g+1)/(3*g^2+g-1) where g = x times the g.f. of A143927. - Mark van Hoeij, Nov 16 2011
a(n) ~ sqrt((221-29*sqrt(13))/78) * ((70+26*sqrt(13))/27)^n/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 25 2014

A102588 Absolute row sums of triangle A102587, which is equal to the matrix inverse of triangle A094531 (the right-hand side of trinomial table A027907).

Original entry on oeis.org

1, 2, 4, 6, 12, 22, 34, 58, 112, 186, 320, 574, 954, 1640, 2926, 5180, 8524, 14928, 25514, 44994, 77674, 138446, 233402, 395832, 704376, 1223902, 2134912, 3628044, 6288414, 10626878, 19171626, 32535060, 57067872, 97164452, 169386950
Offset: 0

Views

Author

Paul D. Hanna, Jan 22 2005

Keywords

Crossrefs

Cf. A094531, A102587.

Programs

  • PARI
    {a(n)=local(A); A=matrix(n+1,n+1,r,c,if(r

A104029 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 13, 9, 1, 5, 26, 35, 14, 1, 6, 45, 96, 75, 20, 1, 7, 71, 216, 267, 140, 27, 1, 8, 105, 427, 750, 623, 238, 35, 1, 9, 148, 770, 1800, 2123, 1288, 378, 44, 1, 10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1, 11, 265, 2067, 7590, 15115, 17303, 11505
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2005

Keywords

Comments

Matrix inverse is A104030. Antidiagonal sums form unsigned A078039.

Examples

			Row 3: {4,13,9,1} is formed from the pairwise sums
of row 3 of A027907: {1,3, 6,7, 6,3, 1}.
Rows begin:
1;
2, 1;
3, 5, 1;
4, 13, 9, 1;
5, 26, 35, 14, 1;
6, 45, 96, 75, 20, 1;
7, 71, 216, 267, 140, 27, 1;
8, 105, 427, 750, 623, 238, 35, 1;
9, 148, 770, 1800, 2123, 1288, 378, 44, 1;
10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1;
11, 265, 2067, 7590, 15115, 17303, 11505, 4302, 825, 65, 1;
12, 341, 3157, 13959, 34210, 49721, 43923, 23397, 7194, 1155, 77, 1; ...

Crossrefs

Cf. A104030, A104027, A078039.

Programs

  • PARI
    {T(n,k)=polcoeff((1+x+x^2)^n+x*O(x^(2*k)),2*k)+ polcoeff((1+x+x^2)^n+x*O(x^(2*k+1)),2*k+1)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
  • PARI
    {T(n,k)=polcoeff(polcoeff((1-x*y)/(1-2*x*(1+y)+x^2*(1+y+y^2)) +x*O(x^n),n,x)+y*O(y^k),k,y)}

Formula

G.f.: A(x, y) = (1-x*y)/(1 - 2*x*(1+y) + x^2*(1+y+y^2) ).
T(n, k) = [x^(2k)](1+x+x^2)^n + [x^(2k+1)](1+x+x^2)^n.

A168591 a(n) = sum of the n-th power of the trinomial coefficients in row n of triangle A027907.

Original entry on oeis.org

1, 3, 19, 831, 281907, 764206503, 16955359883149, 3120135986465328015, 4819828380706011142058787, 63004580363300194268932114354503, 7017256966823394610075464951176481843849
Offset: 0

Views

Author

Paul D. Hanna, Dec 01 2009

Keywords

Crossrefs

Cf. A168590, A027907.

Programs

  • PARI
    a(n)=sum(k=0,2*n,polcoeff((1+x+x^2)^n,k)^n)

Formula

Ignoring initial term, equals the logarithmic derivative of A168590.

A200377 G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..2*n} A027907(n,k)^2 * x^k / A(x)^k) * x^n/n ).

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 19, 34, 61, 106, 181, 311, 543, 955, 1668, 2885, 4980, 8650, 15114, 26391, 45845, 79385, 137718, 239866, 418338, 727926, 1263097, 2191463, 3810775, 6638258, 11556361, 20078960, 34855400, 60567092, 105405431, 183483906, 319039355, 554158992, 962743619, 1674359119, 2913758685, 5068194691
Offset: 0

Views

Author

Paul D. Hanna, Nov 17 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 19*x^6 + 34*x^7 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + x/A + x^2/A^2)*x +
(1 + 2^2*x/A + 3^2*x^2/A^2 + 2^2*x^3/A^3 + x^4/A^4)*x^2/2 +
(1 + 3^2*x/A + 6^2*x^2/A^2 + 7^2*x^3/A^3 + 6^2*x^4/A^4 + 3^2*x^5/A^5 + x^6/A^6)*x^3/3 +
(1 + 4^2*x/A + 10^2*x^2/A^2 + 16^2*x^3/A^3 + 19^2*x^4/A^4 + 16^2*x^5/A^5 + 10^2*x^6/A^6 + 4^2*x^7/A^7 + x^8/A^8)*x^4/4 +
(1 + 5^2*x/A + 15^2*x^2/A^2 + 30^2*x^3/A^3 + 45^2*x^4/A^4 + 51^2*x^5/A^5 + 45^2*x^6/A^6 + 30^2*x^7/A^7 + 15^2*x^8/A^8 + 5^2*x^9/A^9 + x^10/A^10)*x^5/5 +...
which involves the squares of the trinomial coefficients A027907(n,k).

Crossrefs

Cf. A199248, A186236, A199257, A168592, A027907.

Programs

  • PARI
    /* G.f. A(x) using the squares of the trinomial coefficients */
{A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k/(A+x*O(x^n))^k) *x^m/m))); polcoeff(A, n)}
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