A255881 -id:A255881 - cp's OEIS frontend

A255895 O.g.f.: exp( Sum_{n>=1} A000364(n+1)*x^n/n ), where A000364 is the Euler numbers.

1, 5, 43, 635, 15811, 626543, 36670645, 2968583909, 316359049075, 42804896729263, 7156058577755345, 1447112885406569905, 347972935518441794285, 98096607566761160791345, 32036179679728008237066395, 11996707788374286511611298267, 5105803366761640532801681583635

Offset: 0

Paul D. Hanna, Mar 09 2015

nonn

Inspired by Peter Bala's sequence A255881, which suggests the following conjecture: exp( Sum_{n>=1} A000364(n+m)*x^n/n ) is an integer series for integer m>=0.

			O.g.f.: A(x) = 1 + 5*x + 43*x^2 + 635*x^3 + 15811*x^4 + 626543*x^5 +...
where
log(A(x)) = 5*x + 61*x^2/2 + 1385*x^3/3 + 50521*x^4/4 + 2702765*x^5/5 + 199360981*x^6/6 + 19391512145*x^7/7 + 2404879675441*x^8/8 +...+ A000364(n+1)*x^n/n +...

Cf. A255881, A000364.

Table[SeriesCoefficient[Exp[Sum[Abs[EulerE[2*k+2]]*x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2021 *)

/* By the Ordinary Generating Function: */
{A000364(n)=polcoeff(sum(m=0, n, (2*m)!/2^m * x^m/prod(k=1, m, 1+k^2*x+x*O(x^n))), n)}
{a(n)=local(A=1);A=exp(sum(m=1,n,A000364(m+1)*x^m/m) +x*O(x^n));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

a(n) ~ 2^(4*n + 7) * n^(2*n + 3/2) / (exp(2*n) * Pi^(2*n + 5/2)). - Vaclav Kotesovec, Nov 05 2021

A255900 Expansion of exp( Sum_{n >= 1} A000464(n-1)*x^n/n ).

Original entry on oeis.org

1, 1, 6, 126, 6291, 581499, 86010084, 18599726484, 5532984567639, 2166830287921959, 1080602568966548022, 668603866168566179982, 502601850887023486736757, 451167540495577093495790397, 476690705981608679350490956032, 585587125411920596898761442409728

Offset: 0

Peter Bala, Mar 10 2015

For similar results see A188514 and A255881.
A000464(n-1) = (-1)^n*4^(2*n-1)*E(2*n-1,1/4), where E(n,x) denotes the n-th Euler polynomial.
More generally, calculation suggests that for integer h and a nonzero integer k the expansion of exp ( Sum_{n >= 1} (2*k)^(2*n-1)*E(2*n-1,h/(2*k)) )*x^n/n has integer coefficients. This is the case h = 1 and k = 2.

G. C. Greubel, Table of n, a(n) for n = 0..200

Cf. A000464, A188514, A255881.

#A255900
k := 2:
exp(add((2*k)^(2*n-1)*euler(2*n-1, 1/(2*k))*(-x)^n/n, n = 1 .. 15)): seq(coeftayl(%, x = 0, n), n = 0 .. 15);

A000464:= With[{nn = 200}, Take[CoefficientList[Series[Sin[x]/Cos[2 x], {x, 0, nn}], x]*Range[0, nn-1]!, {2, -1, 2}]]; a:= With[{nmax = 50}, CoefficientList[Series[Exp[x + Sum[A000464[[k]]*x^(k)/(k), {k, 2, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( x + 11*x^2/2 + 361*x^3/3 + 24611*x^4/4 + ... ) = 1 + x + 6*x^2 + 126*x^3 + 6291*x^4 + ....

a(0) = 1 and a(n) = 1/n*Sum_{k = 1..n} (-1)^k*4^(2*k-1)*E(2*k-1,1/4)*a(n-k) for n >= 1.

A262143 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} c(n,i)*x^i/i ) for n >= 1, where c(n,k) is Shanks' array of generalized Euler and class numbers.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 8, 33, 23, 1, 16, 208, 1011, 371, 1, 30, 768, 14336, 65985, 10515, 1, 46, 2211, 94208, 2091520, 7536099, 461869, 1, 64, 5043, 412860, 24313856, 535261184, 1329205857, 28969177, 1, 96, 9984, 1361948, 164276421, 11025776640, 211966861312, 334169853267, 2454072147

Offset: 1

Peter Bala, Sep 13 2015

Shanks' array c(n,k) n >= 1, k >= 0, is A235605.
We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 0,1,2,... and for each n >= 1, the expansion of exp( Sum_{i >= 1} c(n,i + r)*x^i/i ) has integer coefficients. The case n = 1 was conjectured by Hanna in A255895.
For the similarly defined array associated with Shanks' d(n,k) array see A262144.

			The square array begins (row indexing n starts at 1)
1  1    3      23        371         10515           461869 ..
1  3   33    1011      65985       7536099       1329205857 ..
1  8  208   14336    2091520     535261184     211966861312 ..
1 16  768   94208   24313856   11025776640    7748875976704 ..
1 30 2211  412860  164276421  115699670490  126686112278631 ..
1 46 5043 1361948  778121381  787337024970 1239870854518999 ..
1 64 9984 3716096 2891509760 3978693525504 8522989918683136 ..
...
Array as a triangle
1
1  1
1  3    3
1  8   33      23
1 16  208    1011      371
1 30  768   14336    65985        10515
1 46 2211   94208  2091520      7536099       461869
1 64 5043  412860  24313856   535261184   1329205857 28969177
1 96 9984 1361948 164276421 11025776640 211966861312 ...
...

P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s) Mathematics of Computation, Vol. 81, No. 278, April 2012.
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Cf. A000233 (column 1), A000364 (c(1,n)), A000281 (c(2,n)), A000436 (c(3,n)), A000490 (c(4,n)), A000187 (c(5,n)), A000192 (c(6,n)), A064068 (c(7,n)), A235605, A235606, A255881, A255895, A262144, A262145.

A383796 Expansion of g.f.: exp(Sum_{n>=1} A295432(n)*x^n/n).

Original entry on oeis.org

1, 462, 396453, 425295010, 511915968714, 661059663660060, 895093835464198893, 1254056426977089876570, 1802794259810040618367902, 2644298823194748929633091780, 3941742074897786728895080586082, 5954164159064906497558129244865108, 9094122817144126105637193154022530612

Offset: 0

Karol A. Penson, Jun 11 2025

nonn

Cf. A295432, A255881, A243953, A166990, A229451.

seq(n)=Vec(exp(sum(n=1, n, (12*n)!*(3*n)!*(2*n)!*x^n/(n*((6*n)!)^2*(4*n)!*n!), O(x*x^n)))) \\ Andrew Howroyd, Jun 11 2025

G.f.: exp(Sum_{n>=1} (12*n)!*(3*n)!*(2*n)!*x^n/(n*((6*n)!)^2*(4*n)!*n!)).

A255912 O.g.f.: exp( Sum_{n>=1} A000364(2n)x^n/n ), where A000364 is the Euler numbers.

Original entry on oeis.org

1, 5, 705, 904405, 4852631105, 74099113400805, 2586129891894933505, 178907219873738420449205, 22190820320340007699602667905, 4580340005051337829651272441809605, 1485137988777113358037521465779043594305, 722514649061693644099760448944719529057242005

Offset: 0

Paul D. Hanna, Mar 10 2015

nonn

a(n) == 5 (mod 100) for n>=1 (conjecture).

			O.g.f.: A(x) = 1 + 5*x + 705*x^2 + 904405*x^3 + 4852631105*x^4 +...
where
log(A(x)) = 5*x + 1385*x^2/2 + 2702765*x^3/3 + 19391512145*x^4/4 + 370371188237525*x^5/5 + 15514534163557086905*x^6/6 + +...+ A000364(2*n)*x^n/n +...

Cf. A255881, A255895, A000364.

/* By definition */
{A000364(n)=polcoeff(sum(m=0, n, (2*m)!/2^m * x^m/prod(k=1, m, 1+k^2*x+x*O(x^n))), n)}
{a(n)=local(A=1);A=exp(sum(m=1,n,A000364(2*m)*x^m/m) +x*O(x^n));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

A381422 Expansion of g.f. = exp( Sum_{n>=1} A066802(n)*x^n/n ).

Original entry on oeis.org

1, 20, 662, 26780, 1205961, 58050204, 2924165436, 152231599628, 8125577046740, 442293253888592, 24457749066666142, 1370114821790970340, 77591333270514869230, 4434803157977731784808, 255492958449660158603448, 14820943641891118200315756, 864962304943085638764540396

Offset: 0

Karol A. Penson, Apr 22 2025

nonn

Cf. A066802, A155200, A255881, A229451, A229452, A156216.

G.f. = 64/((1 + sqrt(1 - 4*x^(1/3)))^2*(1 + sqrt(1 + 4*(-1)^(1/3)*x^(1/3)))^2*(1 + sqrt(1 - 4*(-1)^(2/3)*x^(1/3)))^2).
The above g.f. denoted by h satisfies algebraic equation of order eight:
1 + (8*x - 1)*h + 4*x*(7*x + 3)*h^2 + 7*x^2*(8*x - 1)*h^3 + x^2*(70*x^2 - 40*x + 1)*h^4 + 7*x^4*(8*x - 1)*h^5 + 4*x^5*(7*x + 3)*h^6 + x^6*(8*x - 1)*h^7 + x^8*h^8 = 0.

A384957 Expansion of g.f.: exp(Sum_{n>=1} A295433(n)*x^n/n).

Original entry on oeis.org

1, 990, 2206149, 6450139410, 21553605027306, 77957908218716988, 297118041166459732781, 1175248212459867447863562, 4779368947089383238327733950, 19858241947988743766121587718308, 83936671517628352407663509802203682, 359778601391313651280693986124971038388, 1560159110515342136997114532804454280500084

Offset: 0

Karol A. Penson, Jun 13 2025

nonn

Cf. A295432, A255881, A243953, A166990, A229451, A383796.

seq(n)=Vec(exp(sum(n=1, n, (12*n)!*n!*x^n/((8*n)!*(3*n)!*(2*n)!)/n, O(x*x^n)) )) \\ Andrew Howroyd, Jun 13 2025

G.f.: exp(Sum_{n>=1} (12*n)!*n!*x^n/((8*n)!*(3*n)!*(2*n)!)/n).

cp's OEIS Frontend

A255895 O.g.f.: exp( Sum_{n>=1} A000364(n+1)*x^n/n ), where A000364 is the Euler numbers.

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A255900 Expansion of exp( Sum_{n >= 1} A000464(n-1)*x^n/n ).

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A262143 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} c(n,i)*x^i/i ) for n >= 1, where c(n,k) is Shanks' array of generalized Euler and class numbers.

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A383796 Expansion of g.f.: exp(Sum_{n>=1} A295432(n)*x^n/n).

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A255912 O.g.f.: exp( Sum_{n>=1} A000364(2*n)*x^n/n ), where A000364 is the Euler numbers.

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A381422 Expansion of g.f. = exp( Sum_{n>=1} A066802(n)*x^n/n ).

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A384957 Expansion of g.f.: exp(Sum_{n>=1} A295433(n)*x^n/n).

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A255912 O.g.f.: exp( Sum_{n>=1} A000364(2n)x^n/n ), where A000364 is the Euler numbers.