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 1-10 of 49 results. Next

A159034 Inverse Euler transform of A155200.

Original entry on oeis.org

2, 7, 170, 16380, 6710886, 11453246035, 80421421917330, 2305843009213685760, 268650182136584261045760, 126765060022822940149666965093, 241677817415439249618874010960062650, 1858395433210885261794643189387357732203180, 57560679870263253393868202642364377389525958615670
Offset: 1

Views

Author

Paul D. Hanna, Vladeta Jovovic, Apr 02 2009

Keywords

Links

Crossrefs

Cf. A155200.

Programs

  • Mathematica
    Table[Sum[2^(d^2)*MoebiusMu[n/d], {d, Divisors[n]}]/n, {n, 1, 12}] (* Vaclav Kotesovec, Oct 09 2019 *)
  • PARI
    a(n)={sumdiv(n, d, 2^(d^2)*moebius(n/d))/n} \\ Andrew Howroyd, Jan 08 2020

Formula

a(n) = (1/n)*Sum_{d|n} 2^(d^2)*moebius(n/d).
a(n) ~ 2^(n^2) / n. - Vaclav Kotesovec, Oct 09 2019

Extensions

Terms a(12) and beyond from Andrew Howroyd, Jan 08 2020

A167140 Self-convolution of A155200.

Original entry on oeis.org

1, 4, 24, 416, 34400, 13561728, 22961051392, 160934805885952, 4612329945733989888, 537318814887463743641600, 253532269357851227988228362240, 483356648964255814869226601582346240
Offset: 0

Views

Author

Paul D. Hanna, Oct 30 2009

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 416*x^3 + 34400*x^4 + 13561728*x^5 +...
A(x)^(1/2) = 1 + 2*x + 10*x^2 + 188*x^3 + 16774*x^4 + 6745436*x^5 +...
log(A(x)) = 2^2*x + 2^5*x^2/2 + 2^10*x^3/3 + 2^17*x^4/4 + 2^26*x^5/5 +...

Crossrefs

Cf. A155200.

Programs

  • PARI
    {a(n)=polcoeff(exp( 2*sum(k=1, n, 2^(k^2)*x^k/k)+x*O(x^n)), n)}
  • PARI
    {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, 2^(k^2+1)*a(n-k)))}

Formula

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

A156170 G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^n*x^k]^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 3, 10, 41, 219, 1602, 16635, 247171, 5242108, 157390565, 6663089873, 396778864166, 33200932308437, 3906922702271961, 646161881511137940, 150482521507292513413, 49318093291540113084965, 22790150225552744270503692, 14843990673285561887923674163, 13646527810852572644275538963207, 17710656073227095563348293151121448
Offset: 0

Views

Author

Paul D. Hanna, Feb 05 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 41*x^4 + 219*x^5 + 1602*x^6 +...
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 117*x^4/4 + 821*x^5/5 + 7796*x^6/6 + 1810093*x^7/7 + 44561794*x^8/8 +...+ A276750(n)*x^n/n +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n/n,
or,
log(A(x)) = (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 +...) +
(x + 2^2*x^2 + 3^2*x^3 + 4^2*x^4 + 5^2*x^5 +...)^2/2 +
(x + 2^3*x^2 + 3^3*x^3 + 4^3*x^4 + 5^3*x^5 +...)^3/3 +
(x + 2^4*x^2 + 3^4*x^3 + 4^4*x^4 + 5^4*x^5 +...)^4/4 + ...
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = x/(1-x)^2 + (x + x^2)^2/(1-x)^6/2 + (x + 4*x^2 + x^3)^3/(1-x)^12/3 + (x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^20/4 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^30/5 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^42/6 +...+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n^2+n)/n +...

Links

Crossrefs

Cf. A276750, A276751, A276752, A156171, A155200, A008292, A292500.

Programs

  • PARI
    {a(n) = polcoeff( exp( sum(m=1,n, sum(k=1,n, k^m*x^k +x*O(x^n))^m/m ) ),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {A008292(n,k) = sum(j=0,k, (-1)^j * (k-j)^n * binomial(n+1,j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1,n+1, sum(k=1,m, A008292(m,k)*x^k/(1-x +Oxn)^(m+1) )^m / m ) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: exp( Sum_{n>=1} [ Sum_{k=1..n} A008292(n,k) * x^k / (1-x)^(n+1) ]^n / n ), where A008292 are the Eulerian numbers. - Paul D. Hanna, Sep 13 2016
Conjecture: log(a(n)) ~ n^2 * log(2)/4. - Vaclav Kotesovec, Sep 02 2017

A167006 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) ).

Original entry on oeis.org

1, 2, 6, 66, 4258, 1337374, 1933082159, 11353941470188, 291885138650054688, 29463501750534915665304, 12844314786465829040693498639, 21675661852919288704454219459892060, 156969579902607123047763327413679853875703
Offset: 0

Views

Author

Paul D. Hanna, Nov 17 2009

Keywords

Comments

Logarithmic derivative yields A167009.
Equals row sums of triangle A209196.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 66*x^3 + 4258*x^4 + 1337374*x^5 +...
log(A(x)) = 2*x + 8*x^2/2 + 170*x^3/3 + 16512*x^4/4 + 6643782*x^5/5 + 11582386286*x^6/6 +...+ A167009(n)*x^n/n +...

Links

Crossrefs

Cf. A167009, A209196, A155200.
Cf. variants: A206848, A228809.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m^2,k*m))*x^m/m)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

A155203 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 45, 6687, 10782369, 169490304819, 25016281429306077, 34185693516532070487615, 429210580094546346191627404353, 49269611092414945570325157106493868771
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, for m integer, exp( Sum_{n>=1} m^(n^2) * x^n/n ) is a power series in x with integer coefficients.

Examples

			G.f.: A(x) = 1 + 3*x + 45*x^2 + 6687*x^3 + 10782369*x^4 + 169490304819*x^5 +...
log(A(x)) = 3*x + 3^4*x^2/2 + 3^9*x^3/3 + 3^16*x^4/4 + 3^25*x^5/5 +...

Crossrefs

Cf. A060722, A155204, A155205, A155206, A155812 (triangle), variants: A155200, A155207.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,3^(m^2)*x^m/m)+x*O(x^n)),n)}

Formula

Equals column 0 of triangle A155812.
G.f. satisfies: A'(x)/A(x) = 3 + 27*x*A'(9*x)/A(9*x). - Paul D. Hanna, Nov 15 2022
a(n) ~ 3^(n^2)/n. - Vaclav Kotesovec, Oct 31 2024

A155201 G.f.: A(x) = exp( Sum_{n>=1} (2^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 17, 285, 21747, 7894143, 12593691755, 84961748935779, 2379148487805445513, 273416748863491468927893, 128009274688933686165252807225, 242979449433397149030644307317592609, 1863847996727745781866688849374488247858333, 57652096246331953203644653244501049018464175026133
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, it appears that for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Examples

			G.f.: A(x) = 1 + 3*x + 17*x^2 + 285*x^3 + 21747*x^4 + 7894143*x^5 +...
log(A(x)) = 3*x + 5^2*x^2/2 + 9^3*x^3/3 + 17^4*x^4/4 + 33^5*x^5/5 +...

Crossrefs

Cf. A136516, A155200, A155202, A155810 (triangle), variants: A155204, A155208.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,(2^m+1)^m*x^m/m)+x*O(x^n)),n)}

Formula

Equals row sums of triangle A155810.
a(n) = (1/n)*Sum_{k=1..n} (2^k + 1)^k * a(n-k) for n>0, with a(0)=1.
a(n) = B_n( 0!*(2^1+1)^1, 1!*(2^2+1)^2, 2!*(2^3+1)^3, ..., (n-1)!*(2^n+1)^n ) / n!, where B_n() is the n-th complete Bell polynomial. - Max Alekseyev, Oct 10 2014

A209196 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * y^k ), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 32, 32, 1, 1, 487, 3282, 487, 1, 1, 11113, 657573, 657573, 11113, 1, 1, 335745, 209282906, 1513844855, 209282906, 335745, 1, 1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1, 1, 565877928, 61162554558200
Offset: 0

Views

Author

Paul D. Hanna, Mar 05 2012

Keywords

Examples

			This triangle begins:
1;
1, 1;
1, 4, 1;
1, 32, 32, 1;
1, 487, 3282, 487, 1;
1, 11113, 657573, 657573, 11113, 1;
1, 335745, 209282906, 1513844855, 209282906, 335745, 1;
1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1;
1, 565877928, 61162554558200, 31336815578461815, 229089181252258800, 31336815578461815, 61162554558200, 565877928, 1; ...
G.f.: A(x,y) = 1 + (1+y)*x + (1+4*y+y^2)*x^2 + (1+32*y+32*y^2+y^3)*x^3 + (1+487*y+3282*y^2+487*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 6*y + y^2)*x^2/2
+ (1 + 84*y + 84*y^2 + y^3)*x^3/3
+ (1 + 1820*y + 12870*y^2 + 1820*y^3 + y^4)*x^4/4
+ (1 + 53130*y + 3268760*y^2 + 3268760*y^3 + 53130*y^4 + y^5)*x^5/5 +...
in which the coefficients form A209330(n,k) = binomial(n^2, n*k).

Links

Crossrefs

Cf. A209197, A167006, A206830, A209330 (log), A155200.

Programs

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

Formula

Column 1 equals A209197.
Row sums equal A167006.
Antidiagonal sums equal A206830.

A155202 G.f.: A(x) = exp( Sum_{n>=1} (2^n - 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 5, 119, 12783, 5739069, 10426379903, 76135573607705, 2234839096465512877, 263966776643953756165279, 125532809982533901346598445525, 240383033223427436734891985275952307
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 119*x^3 + 12783*x^4 + 5739069*x^5 +...
log(A(x)) = x + 3^2*x^2/2 + 7^3*x^3/3 + 15^4*x^4/4 + 31^5*x^5/5 +...

Crossrefs

Cf. A055601, A155200, A155202, A155810 (triangle), variants: A155205, A155209.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,(2^m-1)^m*x^m/m)+x*O(x^n)),n)}

A155207 G.f.: A(x) = exp( Sum_{n>=1} 4^(n^2) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 136, 87904, 1074100576, 225184288253824, 787061981348092400896, 45273238870711805132010916864, 42535296046210357883346895894694749696, 649556283428320264374891976653586736162144180224
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, for m integer, exp( Sum_{n>=1} m^(n^2) * x^n/n ) is a power series in x with integer coefficients.

Examples

			G.f.: A(x) = 1 + 4*x + 136*x^2 + 87904*x^3 + 1074100576*x^4 +...
log(A(x)) = 4*x + 4^4*x^2/2 + 4^9*x^3/3 + 4^16*x^4/4 + 4^25*x^5/5 +...

Crossrefs

Cf. A060757, A155208, A155209, A155210, variants: A155200, A155203.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,4^(m^2)*x^m/m)+x*O(x^n)),n)}

Formula

G.f. satisfies: A'(x)/A(x) = 4 + 64*x*A'(16*x)/A(16*x). - Paul D. Hanna, Nov 15 2022
a(n) ~ 4^(n^2)/n. - Vaclav Kotesovec, Oct 31 2024

A155810 Triangle, read by rows, where g.f.: A(x,y) = exp( Sum_{n>=1} (2^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Original entry on oeis.org

1, 2, 1, 10, 6, 1, 188, 82, 14, 1, 16774, 4452, 490, 30, 1, 6745436, 1074934, 71108, 2602, 62, 1, 11466849412, 1082704500, 43173414, 951300, 13002, 126, 1, 80444398636280, 4411700155252, 104251164804, 1387446246, 11470404, 62538, 254, 1, 2306003967992402758, 72146891831948808, 989785148972932, 7803708940836, 38993810694, 129076164, 292810, 510, 1, 268654794629082985019564, 4724816968764733073446, 36967624172237518088, 169140002768370820, 500466007443108, 1001353593606, 1382564804, 1343434, 1022, 1
Offset: 0

Views

Author

Paul D. Hanna, Feb 04 2009

Keywords

Comments

More generally, for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Examples

			G.f.: A(x,y) = 1 + (2 + y)x + (10 + 6y + y^2)x^2 + (188 + 82y + 14y^2 + y^3)x^3 +...
Triangle begins:
1;
2, 1;
10, 6, 1;
188, 82, 14, 1;
16774, 4452, 490, 30, 1;
6745436, 1074934, 71108, 2602, 62, 1;
11466849412, 1082704500, 43173414, 951300, 13002, 126, 1;
80444398636280, 4411700155252, 104251164804, 1387446246, 11470404, 62538, 254, 1;
2306003967992402758, 72146891831948808, 989785148972932, 7803708940836, 38993810694, 129076164, 292810, 510, 1;
268654794629082985019564, 4724816968764733073446, 36967624172237518088, 169140002768370820, 500466007443108, 1001353593606, 1382564804, 1343434, 1022, 1; ...

Crossrefs

Cf. A155200 (column 0), A155201 (row sums), A155811 (column 1).

Programs

  • PARI
    {T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n+1,(2^m+y)^m*x^m/m)+x*O(x^n)),n,x),k,y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Formula

G.f.: A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k.
Showing 1-10 of 49 results. Next

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