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.

A245373 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 2*(n+1)*x) ).

Original entry on oeis.org

1, 2, 6, 20, 80, 368, 1904, 10880, 67904, 459008, 3336704, 25925120, 214175744, 1873092608, 17276401664, 167504076800, 1702214549504, 18084149854208, 200388963958784, 2311212530401280, 27693720143396864, 344157474490155008, 4428851361694613504, 58933575269230837760
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2014

Keywords

Comments

A generalization of Peter Bala's formula in A229046 is as follows:
if F(x,y) = Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - (n+1)*x*y) ) then
F(x,y) = Sum_{n>=0} n! * (x*y)^n * (1+x)^n / Product_{k=1..n} (1 + k*x*y);
further, F(x,y) = Sum_{n>=0} b(n,y)*x^n where b(n,y) is given by
b(n,y) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * C(k,i) * (k-i+1)^(n-k) * y^(n-k).

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 80*x^4 + 368*x^5 + 1904*x^6 +...
where we have the following series identity:
A(x) = 1/((1+x)*(1-2*x)) + x/((1+x)^2*(1-4*x)) + x^2/((1+x)^3*(1-6*x))+ x^3/((1+x)^4*(1-8*x))+ x^4/((1+x)^5*(1-10*x)) + x^5/((1+x)^6*(1-12*x)) +...
is equal to
A(x) = 1 + 2*x*(1+x)/(1+2*x) + 2!*(2*x)^2*(1+x)^2/((1+2*x)*(1+4*x)) + 3!*(2*x)^3*(1+x)^3/((1+2*x)*(1+4*x)*(1+6*x)) + 4!*(2*x)^4*(1+x)^4/((1+2*x)*(1+4*x)*(1+6*x)*(1+8*x)) + 5!*(2*x)^5*(1+x)^5/((1+2*x)*(1+4*x)*(1+6*x)*(1+8*x)*(1+10*x)) +...

Crossrefs

Cf. A229046, A245374, A245375, A245376.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 2*(m+1)*x) +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff( sum(m=0, n, 2^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+2*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, floor(n/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k)*2^(n-k)))}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} n! * (2*x)^n * (1+x)^n / Product_{k=1..n} (1 + 2*k*x).
a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * 2^(n-k) * (k-i+1)^(n-k).

A245374 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 3*(n+1)*x) ).

Original entry on oeis.org

1, 3, 12, 54, 288, 1782, 12474, 96714, 819882, 7536402, 74610234, 790692354, 8921660922, 106687646802, 1346863560714, 17890362862434, 249297686894682, 3634756665823602, 55317506662094634, 876911386062810114, 14451743847813157242, 247171758180997987602, 4380263376360686471754
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2014

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 54*x^3 + 288*x^4 + 1782*x^5 + 12474*x^6 +...
where we have the following series identity:
A(x) = 1/((1+x)*(1-3*x)) + x/((1+x)^2*(1-6*x)) + x^2/((1+x)^3*(1-9*x))+ x^3/((1+x)^4*(1-12*x))+ x^4/((1+x)^5*(1-15*x)) + x^5/((1+x)^6*(1-18*x)) +...
is equal to
A(x) = 1 + 3*x*(1+x)/(1+3*x) + 2!*(3*x)^2*(1+x)^2/((1+3*x)*(1+6*x)) + 3!*(3*x)^3*(1+x)^3/((1+3*x)*(1+6*x)*(1+9*x)) + 4!*(3*x)^4*(1+x)^4/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)) + 5!*(3*x)^5*(1+x)^5/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)*(1+15*x)) +...

Crossrefs

Cf. A229046, A245373, A245375, A245376.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 3*(m+1)*x) +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff( sum(m=0, n, 3^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+3*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, floor(n/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k)*3^(n-k)))}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} n! * (3*x)^n * (1+x)^n / Product_{k=1..n} (1 + 3*k*x).
a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * 3^(n-k) * (k-i+1)^(n-k).

A245376 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 5*(n+1)*x) ).

Original entry on oeis.org

1, 5, 30, 200, 1550, 14000, 144500, 1662500, 20952500, 286437500, 4221312500, 66703437500, 1124194062500, 20109785937500, 380209901562500, 7571141773437500, 158312671414062500, 3466819503710937500, 79316483272226562500, 1891747084452148437500, 46942864023040039062500
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2014

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 30*x^2 + 200*x^3 + 1550*x^4 + 14000*x^5 +...
where we have the following series identity:
A(x) = 1/((1+x)*(1-5*x)) + x/((1+x)^2*(1-10*x)) + x^2/((1+x)^3*(1-15*x))+ x^3/((1+x)^4*(1-20*x))+ x^4/((1+x)^5*(1-25*x)) + x^5/((1+x)^6*(1-30*x)) +...
is equal to
A(x) = 1 + 5*x*(1+x)/(1+5*x) + 2!*(5*x)^2*(1+x)^2/((1+5*x)*(1+10*x)) + 3!*(5*x)^3*(1+x)^3/((1+5*x)*(1+10*x)*(1+15*x)) + 4!*(5*x)^4*(1+x)^4/((1+5*x)*(1+10*x)*(1+15*x)*(1+20*x)) + 5!*(5*x)^5*(1+x)^5/((1+5*x)*(1+10*x)*(1+15*x)*(1+20*x)*(1+25*x)) +...

Crossrefs

Cf. A229046, A245373, A245374, A245375.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 5*(m+1)*x) +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff( sum(m=0, n, 5^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+5*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, floor(n/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k)*5^(n-k)))}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} n! * (5*x)^n * (1+x)^n / Product_{k=1..n} (1 + 5*k*x).
a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * 5^(n-k) * (k-i+1)^(n-k).

A245378 G.f. satisfies: A(x) = Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - (n+1)*x*A(x)) ).

Original entry on oeis.org

1, 1, 3, 10, 39, 165, 743, 3507, 17199, 87126, 454159, 2430031, 13326623, 74856230, 430628069, 2538270783, 15343363603, 95233568052, 607850790015, 3996223189468, 27105153736781, 189947851239185, 1376864409041417, 10330672337146804, 80248762443834399, 645206035074873681
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2014

Keywords

Comments

Compare g.f. to an identity for C(x) = 1 + x*C(x)^2, the Catalan function:
C(x) = Sum_{n>=0} n! * x^n * C(x)^n / Product_{k=1..n} (1 + k*x*C(x)).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 165*x^5 + 743*x^6 +...
where we have the following identity:
A(x) = 1/((1+x)*(1-x*A(x))) + x/((1+x)^2*(1-2*x*A(x))) + x^2/((1+x)^3*(1-3*x*A(x)))+ x^3/((1+x)^4*(1-4*x*A(x)))+ x^4/((1+x)^5*(1-5*x*A(x))) + x^5/((1+x)^6*(1-6*x*A(x))) +...
is equal to
A(x) = 1 + x*A(x)*(1+x)/(1+x*A(x)) + 2!*x^2*A(x)^2*(1+x)^2/((1+x*A(x))*(1+2*x*A(x))) + 3!*x^3*A(x)^3*(1+x)^3/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))) + 4!*x^4*A(x)^4*(1+x)^4/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))) + 5!*x^5*A(x)^5*(1+x)^5/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))*(1+5*x*A(x))) +...

Links

Crossrefs

Cf. A229046, A245373, A245374, A245375, A245376.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, x^m/((1+x)^(m+1)*(1 - (m+1)*x*A +x*O(x^n)))));polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0, n, m!*x^m*A^m*(1+x)^m/prod(k=1, m, 1+k*x*A +x*O(x^n))));polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^n * (1+x)^n / Product_{k=1..n} (1 + k*x*A(x)).
Showing 1-4 of 4 results.

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