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 1-3 of 3 results.

A227313 Sum of squared terms in rows of triangle A227311.

Original entry on oeis.org

4, 16, 64, 320, 1424, 6400, 28928, 131328, 599428, 2746176, 12627520, 58237952, 269284240, 1247921152, 5794490624, 26952340480, 125559053904, 585733448080, 2735853906496, 12793091964160, 59883014554112, 280568427766016, 1315670787139840, 6174463935221760, 28998036439469524, 136279914514165568, 640867368366269056
Offset: 1

Views

Author

Paul D. Hanna, Jul 06 2013

Keywords

Examples

			L.g.f.: L(x) = 4*x + 16*x^2/2 + 64*x^3/3 + 320*x^4/4 + 1424*x^5/5 + 6400*x^6/6 +...
where
exp(L(x)) = 1 + 4*x + 16*x^2 + 64*x^3 + 272*x^4 + 1168*x^5 + 5056*x^6 + 22016*x^7 +...

Crossrefs

Cf. A227311.

Programs

  • PARI
    {A227311(n,k)=n*polcoeff(polcoeff(log(1 + 2*sum(m=1,sqrtint(n),y^m*x^(m^2))+x*O(x^n)),n,x),k,y)}
{a(n)=sum(k=1,n,A227311(n,k)^2)}
for(n=1, 36, print1(a(n), ", "))

A216273 Triangle generated by Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k/n = log(1 + Sum_{n>=1} y*x^(n^2)), where coefficients are read by rows.

Original entry on oeis.org

1, 0, -1, 0, 0, 1, 4, 0, 0, -1, 0, -5, 0, 0, 1, 0, 0, 6, 0, 0, -1, 0, 0, 0, -7, 0, 0, 1, 0, -4, 0, 0, 8, 0, 0, -1, 9, 0, 9, 0, 0, -9, 0, 0, 1, 0, -10, 0, -15, 0, 0, 10, 0, 0, -1, 0, 0, 11, 0, 22, 0, 0, -11, 0, 0, 1, 0, 0, 4, -12, 0, -30, 0, 0, 12, 0, 0, -1, 0, -13, 0, -13, 13, 0, 39, 0, 0, -13, 0, 0, 1, 0, 0, 28, 0, 28, -14, 0, -49, 0, 0, 14, 0, 0, -1
Offset: 1

Views

Author

Paul D. Hanna, Mar 16 2013

Keywords

Examples

			G.f.: A(x,y) = y*x - y^2*x^2/2 + y^3*x^3/3 + (-y^4 + 4*y)*x^4/4 + (y^5 - 5*y^2)*x^5/5 + (-y^6 + 6*y^3)*x^6/6 + (y^7 - 7*y^4)*x^7/7 + (-y^8 + 8
*y^5 - 4*y^2)*x^8/8 + (y^9 - 9*y^6 + 9*y^3 + 9*y)*x^9/9 + (-y^10 + 10*y^7 - 15*y^4 - 10*y^2)*x^10/10 +...
where
exp(A(x,y)) = 1 + y*x + y*x^4 + y*x^9 + y*x^16 + y*x^25 +...
Triangle begins:
1;
0, -1;
0, 0, 1;
4, 0, 0, -1;
0, -5, 0, 0, 1;
0, 0, 6, 0, 0, -1;
0, 0, 0, -7, 0, 0, 1;
0, -4, 0, 0, 8, 0, 0, -1;
9, 0, 9, 0, 0, -9, 0, 0, 1;
0, -10, 0, -15, 0, 0, 10, 0, 0, -1;
0, 0, 11, 0, 22, 0, 0, -11, 0, 0, 1;
0, 0, 4, -12, 0, -30, 0, 0, 12, 0, 0, -1;
0, -13, 0, -13, 13, 0, 39, 0, 0, -13, 0, 0, 1;
0, 0, 28, 0, 28, -14, 0, -49, 0, 0, 14, 0, 0, -1;
0, 0, 0, -45, 0, -50, 15, 0, 60, 0, 0, -15, 0, 0, 1;
16, 0, 0, -4, 64, 0, 80, -16, 0, -72, 0, 0, 16, 0, 0, -1;
0, -17, 17, 0, 17, -85, 0, -119, 17, 0, 85, 0, 0, -17, 0, 0, 1;
0, -9, 18, -54, 0, -45, 108, 0, 168, -18, 0, -99, 0, 0, 18, 0, 0, -1;
0, 0, 19, -19, 114, 0, 95, -133, 0, -228, 19, 0, 114, 0, 0, -19, 0, 0, 1;
0, -20, 0, -30, 24, -200, 0, -175, 160, 0, 300, -20, 0, -130, 0, 0, 20, 0, 0, -1;
0, 0, 42, -21, 42, -42, 315, 0, 294, -189, 0, -385, 21, 0, 147, 0, 0, -21, 0, 0, 1;
0, 0, 22, -66, 88, -55, 88, -462, 0, -462, 220, 0, 484, -22, 0, -165, 0, 0, 22, 0, 0, -1;
0, 0, 0, -69, 92, -230, 69, -184, 644, 0, 690, -253, 0, -598, 23, 0, 184, 0, 0, -23, 0, 0, 1;
0, 0, 24, 0, 144, -124, 480, -84, 360, -864, 0, -990, 288, 0, 728, -24, 0, -204, 0, 0, 24, 0, 0, -1;
25, -25, 0, -75, 25, -250, 175, -875, 100, -655, 1125, 0, 1375, -325, 0, -875, 25, 0, 225, 0, 0, -25, 0, 0, 1; ...

Links

Crossrefs

Cf. A227311, A162552, A000203, A015128, A002318, A004404, A004405.

Programs

  • PARI
    {T(n,k)=n*polcoeff(polcoeff(log(1+sum(m=1,sqrtint(n)+1,y*x^(m^2))+x*O(x^n)),n,x),k,y)}
for(n=1,25,for(k=1,n,print1(T(n,k),", "));print(""))
  • PARI
    /* Alternate g.f., true for all m >= 0: */
{T(n,k,m=0) = if(k<1||m<0,0, (n/k/binomial(k+m,m)) * polcoeff(polcoeff( 1 - 1/(1+sum(j=1,sqrtint(n+1),y*x^(j^2))+x*O(x^n))^(m+1), n,x),k,y))}
for(n=1, 25, for(k=1, n, print1(T(n, k, 1), ", ")); print(""))

Formula

G.f.: Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k*k*binomial(k+m,m)/n = 1 - 1/(1 + Sum_{n>=1} y*x^(n^2))^(m+1), which holds for all m >= 0.
Row sums equal A162552.
Sum_{k=1..n} T(n,k)*2^k = -(-1)^n*(sigma(2*n) - sigma(n)) for n>=1, where sigma is the sum of divisors of n, A000203.
Sum_{k=1..n} T(n,k)*2^k*k = -(-1)^n*n*A015128(n) for n>=1, where A015128(n) is the number of overpartitions of n, with g.f.: Product_{n>=1} (1+x^n)/(1-x^n).
Sum_{k=1..n} T(n,k)*2^k*k*(k+1) = -(-1)^n*4*n*A002318(n) for n>=1, where A002318 lists the coefficients in (1/theta_4(q)^2 -1)/4 in powers of q.
Sum_{k=1..n} T(n,k)*2^k*k*(k+1)*(k+2)/2! = -n*A004404(n) for n>=1, where A004404 lists the coefficients in 1/(1 + Sum_{n>=1} 2*x^(n^2))^3.
Sum_{k=1..n} T(n,k)*2^k*k*(k+1)*(k+2)*(k+3)/3! = -n*A004405(n) for n>=1, where A004405 lists the coefficients in 1/(1 + Sum_{n>=1} 2*x^(n^2))^4.
More generally:
Sum_{k=1..n} T(n,k)*y^k*k*binomial(k+m,m)/n = [x^n] 1 - 1/(1 + Sum_{n>=1} y*x^(n^2))^(m+1) for m>=0, n>=1.

A227312 L.g.f.: log(1 + 2*Sum_{n>=1} 2^n * x^(n^2)).

Original entry on oeis.org

4, -16, 64, -224, 864, -3328, 12800, -49408, 190864, -736896, 2845440, -10987520, 42426752, -163825664, 632592384, -2442673664, 9432071040, -36420732160, 140633977856, -543040041984, 2096879372288, -8096830353408, 31264870391808, -120725281128448, 466165166208064, -1800036911561216, 6950611323771904
Offset: 1

Views

Author

Paul D. Hanna, Jul 06 2013

Keywords

Comments

Compare to the logarithm of theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2):
log(theta_3(x)) = Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n, where sigma is the sum of divisors of n (A000203).

Examples

			L.g.f.: L(x) = 4*x - 16*x^2/2 + 64*x^3/3 - 224*x^4/4 + 864*x^5/5 - 3328*x^6/6 + 12800*x^7/7 - 49408*x^8/8 + 190864*x^9/9 - 736896*x^10/10 + 2845440*x^11/11 - 10987520*x^12/12 + 42426752*x^13/13 - 163825664*x^14/14 + 632592384*x^15/15 - 2442673664*x^16/16 +...
where
exp(L(x)) = 1 + 4*x + 8*x^4 + 16*x^9 + 32*x^16 + 64*x^25 + 128*x^36 + 256*x^49 +...

Crossrefs

Cf. A227311, A227313, A227315.

Programs

  • PARI
    {a(n)=n*polcoeff(log(1+sum(k=1, n, 2*2^k*x^(k^2))+x*O(x^n)), n)}
for(n=1, 36, print1(a(n), ", "))
Showing 1-3 of 3 results.

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