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Search: a155200 -id:a155200
Displaying 1-10 of 48 results found. page 1 2 3 4 5
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A159034 Inverse Euler transform of A155200. +20
2
2, 7, 170, 16380, 6710886, 11453246035, 80421421917330, 2305843009213685760, 268650182136584261045760, 126765060022822940149666965093, 241677817415439249618874010960062650, 1858395433210885261794643189387357732203180, 57560679870263253393868202642364377389525958615670 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..50
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
MATHEMATICA
Table[Sum[2^(d^2)*MoebiusMu[n/d], {d, Divisors[n]}]/n, {n, 1, 12}] (* Vaclav Kotesovec, Oct 09 2019 *)
PROG
(PARI) a(n)={sumdiv(n, d, 2^(d^2)*moebius(n/d))/n} \\ Andrew Howroyd, Jan 08 2020
CROSSREFS
Cf. A155200.
KEYWORD
easy,nonn
AUTHOR
Paul D. Hanna, Vladeta Jovovic, Apr 02 2009
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Jan 08 2020
STATUS
approved
A167140 Self-convolution of A155200. +20
0
1, 4, 24, 416, 34400, 13561728, 22961051392, 160934805885952, 4612329945733989888, 537318814887463743641600, 253532269357851227988228362240, 483356648964255814869226601582346240 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..11.
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.
EXAMPLE
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 +...
PROG
(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)))}
CROSSREFS
Cf. A155200.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 30 2009
STATUS
approved
A156170 G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^n*x^k]^n/n ), a power series in x with integer coefficients. +10
15
1, 1, 3, 10, 41, 219, 1602, 16635, 247171, 5242108, 157390565, 6663089873, 396778864166, 33200932308437, 3906922702271961, 646161881511137940, 150482521507292513413, 49318093291540113084965, 22790150225552744270503692, 14843990673285561887923674163, 13646527810852572644275538963207, 17710656073227095563348293151121448 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..150
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
EXAMPLE
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 +...
PROG
(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), ", "))
CROSSREFS
Cf. A276750, A276751, A276752, A156171, A155200, A008292, A292500.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 05 2009
STATUS
approved
A167006 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) ). +10
13
1, 2, 6, 66, 4258, 1337374, 1933082159, 11353941470188, 291885138650054688, 29463501750534915665304, 12844314786465829040693498639, 21675661852919288704454219459892060, 156969579902607123047763327413679853875703 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Logarithmic derivative yields A167009.
Equals row sums of triangle A209196.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..57
EXAMPLE
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 +...
PROG
(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), ", "))
CROSSREFS
Cf. A167009, A209196, A155200.
Cf. variants: A206848, A228809.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 17 2009
STATUS
approved
A155203 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2) * x^n/n ), a power series in x with integer coefficients. +10
11
1, 3, 45, 6687, 10782369, 169490304819, 25016281429306077, 34185693516532070487615, 429210580094546346191627404353, 49269611092414945570325157106493868771 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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.
LINKS
Table of n, a(n) for n=0..9.
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
EXAMPLE
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 +...
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, 3^(m^2)*x^m/m)+x*O(x^n)), n)}
CROSSREFS
Cf. A155204, A155205, A155206, A155812 (triangle), variants: A155200, A155207.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2009
STATUS
approved
A155201 G.f.: A(x) = exp( Sum_{n>=1} (2^n + 1)^n * x^n/n ), a power series in x with integer coefficients. +10
8
1, 3, 17, 285, 21747, 7894143, 12593691755, 84961748935779, 2379148487805445513, 273416748863491468927893, 128009274688933686165252807225, 242979449433397149030644307317592609, 1863847996727745781866688849374488247858333, 57652096246331953203644653244501049018464175026133 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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.
LINKS
Table of n, a(n) for n=0..13.
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
EXAMPLE
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 +...
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, (2^m+1)^m*x^m/m)+x*O(x^n)), n)}
CROSSREFS
Cf. A155200, A155202, A155810 (triangle), variants: A155204, A155208.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2009
STATUS
approved
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. +10
8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Paul D. Hanna, Rows n = 0..30, flattened.
FORMULA
Column 1 equals A209197.
Row sums equal A167006.
Antidiagonal sums equal A206830.
EXAMPLE
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).
PROG
(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(""))
CROSSREFS
Cf. A209197, A167006, A206830, A209330 (log), A155200.
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 05 2012
STATUS
approved
A155202 G.f.: A(x) = exp( Sum_{n>=1} (2^n - 1)^n * x^n/n ), a power series in x with integer coefficients. +10
7
1, 1, 5, 119, 12783, 5739069, 10426379903, 76135573607705, 2234839096465512877, 263966776643953756165279, 125532809982533901346598445525, 240383033223427436734891985275952307 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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.
LINKS
Table of n, a(n) for n=0..11.
EXAMPLE
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 +...
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, (2^m-1)^m*x^m/m)+x*O(x^n)), n)}
CROSSREFS
Cf. A155200, A155202, A155810 (triangle), variants: A155205, A155209.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2009
STATUS
approved
A155207 G.f.: A(x) = exp( Sum_{n>=1} 4^(n^2) * x^n/n ), a power series in x with integer coefficients. +10
6
1, 4, 136, 87904, 1074100576, 225184288253824, 787061981348092400896, 45273238870711805132010916864, 42535296046210357883346895894694749696, 649556283428320264374891976653586736162144180224 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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.
LINKS
Table of n, a(n) for n=0..9.
FORMULA
G.f. satisfies: A'(x)/A(x) = 4 + 64*x*A'(16*x)/A(16*x). - Paul D. Hanna, Nov 15 2022
EXAMPLE
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 +...
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, 4^(m^2)*x^m/m)+x*O(x^n)), n)}
CROSSREFS
Cf. A155208, A155209, A155210, variants: A155200, A155203.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2009
STATUS
approved
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. +10
6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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.
LINKS
Table of n, a(n) for n=0..54.
FORMULA
G.f.: A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k.
EXAMPLE
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; ...
PROG
(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(""))
CROSSREFS
Cf. A155200 (column 0), A155201 (row sums), A155811 (column 1).
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Feb 04 2009
STATUS
approved
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Last modified August 29 22:07 EDT 2024. Contains 375518 sequences. (Running on oeis4.)