A139757 -id:A139757 - cp's OEIS frontend

A015223 Odd pentagonal pyramidal numbers.

1, 75, 405, 1183, 2601, 4851, 8125, 12615, 18513, 26011, 35301, 46575, 60025, 75843, 94221, 115351, 139425, 166635, 197173, 231231, 269001, 310675, 356445, 406503, 461041, 520251, 584325, 653455, 727833, 807651, 893101, 984375

Offset: 0

Mohammad K. Azarian

Also first bisection of A139757. - Bruno Berselli, Feb 13 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002411, A014799, A015224, A014800.

[(2*n+1)*(4*n+1)^2: n in [0..50]]; // G. C. Greubel, Nov 04 2017

Table[((n+1)^3+(n+1)^2)/2,{n,0,200,4}] (* Vladimir Joseph Stephan Orlovsky, May 21 2011 *)
CoefficientList[Series[(1 + 71 x + 111 x^2 + 9 x^3)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 15 2013 *)

a(n)=(2*n+1)*(4*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1 + 71*x + 111*x^2 + 9*x^3)/(1-x)^4. - Colin Barker, Feb 13 2012
a(n) = (2n+1)*(4n+1)^2 = A130656(4n+1). - Bruno Berselli, Feb 13 2012
From Ant King, Oct 23 2012: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 192.
Sum_{n>=0} 1/a(n) = (8*C - 2*Pi + Pi^2 - 4*log(2))/8, where C is Catalan's constant (A006752). (End)
E.g.f.: (1 + 74*x + 128*x^2 + 32*x^3)*exp(x). - G. C. Greubel, Nov 04 2017

More terms from Erich Friedman

A144849 Coefficients in the expansion of the squared sine lemniscate function.

Original entry on oeis.org

1, 6, 336, 77616, 50916096, 76307083776, 226653840838656, 1207012936807028736, 10696277678308486742016, 148900090457044541209706496, 3110043187741674836967136690176, 93885206124269301790338015801901056, 3970859549814416912519992571903015387136

Offset: 0

N. J. A. Sloane, Feb 12 2009

nonn

Denoted by \beta_n in Lomont and Brillhart (2011) on page xiii.

Gives the number of Increasing bilabeled strict binary trees with 4n+2 labels. - Markus Kuba, Nov 18 2014

			G.f. = 1 + 6*x + 336*x^2 + 77616*x^3 + 50916096*x^4 + ...

J. S. Lomont and J. Brillhart, Elliptic Polynomials, Chapman and Hall, 2001; see p. 86.

N. J. A. Sloane, Table of n, a(n) for n = 0..100
O. Bodini, M. Dien, X. Fontaine, A. Genitrini, and H. K. Hwang, Increasing Diamonds, in LATIN 2016: 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings Pages pp 207-219 2016 DOI 10.1007/978-3-662-49529-2_16; Lecture Notes in Computer Science Series Volume 9644.
Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], (17-November-2014).
Tanay Wakhare, Christophe Vignat, Taylor coefficients of the Jacobi theta3(q) function, arXiv:1909.01508 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Lemniscate Constant

Cf. A064853, A144853.

a[0]:=1; b[0]:=1;
for n from 1 to 15 do b[n]:=add(binomial(4*n,4*j+2)*b[j]*b[n-1-j],j=0..n-1);
a[n]:=(1/3)*add(binomial(4*n-1,4*j+1)*a[j]*b[n-1-j],j=0..n-1); od:
tb:=[seq(b[n],n=0..15)];

a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 2}, m! SeriesCoefficient[ JacobiSD[ x, 1/2]^2, {x, 0, m}] / (2 (-3)^n)]]; (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 2}, m! SeriesCoefficient[ InverseSeries[ Integrate[ Series[ (1 + x^4 / 12) ^ (-1/2), {x, 0, m + 1}], x]]^2 / 2, {x, 0, m}]]]; (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[ n < 1, Boole[n == 0], Sum[ Binomial[ 4 n, 4 j + 2] a[j] a[ n - 1 - j], {j, 0, n - 1}]]; (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[n < 0, 0, With[{m = 4*n + 2}, m!*SeriesCoefficient[JacobiSN[x, -1]^2, {x, 0, m}]/(2*(-12)^n)]]; (* Michael Somos, Jul 10 2024 *)

{a(n) = my(m); if( n<0, 0, m = 4*n + 2; m! * polcoeff( (serreverse( intformal( (1 + x^4 / 12 + x * O(x^m)) ^ (-1/2))))^2 / 2, m))}; /* Michael Somos, Apr 25 2011 */

E.g.f.: sl(x)^2 = 2 Sum_{k>=0} (-12)^k * a(k) * x^(4*k + 2) / (4*k + 2)! where sl(x) = sin lemn(x) is the sine lemniscate function of Gauss. - Michael Somos, Apr 25 2011
a(0) = 1, a(n + 1) = Sum_{j=0..n} binomial( 4*n + 4, 4*j + 2) * a(j) * a(n - j).
G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b(n) = A139757(n) * n/3. - Michael Somos, Jan 03 2013
E.g.f.: Increasing bilabeled strict binary trees of 2n+2 labels (including the zeros): T(z)=Sum_{n>=1}T_n z^{2n}/(2n)! = 6/sqrt(3)*WeierstrassP(3^{-1/4}z+LemniscateConstant; g_2,g_3), with g_2=-1 and g_3=0; alternatively, T(z)=sqrt(3)*i*sl^2(z/(3^{1/4}(1+i))). - Markus Kuba, Nov 18 2014

A329530 a(n) = n * (7*binomial(n, 2) + 1).

Original entry on oeis.org

0, 1, 16, 66, 172, 355, 636, 1036, 1576, 2277, 3160, 4246, 5556, 7111, 8932, 11040, 13456, 16201, 19296, 22762, 26620, 30891, 35596, 40756, 46392, 52525, 59176, 66366, 74116, 82447, 91380, 100936, 111136, 122001, 133552, 145810, 158796, 172531, 187036, 202332, 218440

Offset: 0

Ilya Gutkovskiy, Nov 15 2019

Centered heptagonal prism numbers.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), 144.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Centered m-gonal prism numbers: A100175 (m = 3), A059722 (m = 4), A006564 (m = 5), A005915 (m = 6), this sequence (m = 7), A139757 (m = 8), A006566 (m = 9).

Cf. A000217, A002413, A006597, A024966, A069099.

Table[n (7 Binomial[n, 2] + 1), {n, 0, 40}]
nmax = 40; CoefficientList[Series[x (1 + 12 x + 8 x^2)/(1 - x)^4, {x, 0, nmax}], x]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 16, 66}, 41]

G.f.: x * (1 + 12*x + 8*x^2) / (1 - x)^4.
E.g.f.: exp(x) * x * (2 + 14*x + 7*x^2) / 2.
a(n) = n * (7*n^2 - 7*n + 2) / 2.
a(n) = n * (7*A000217(n-1) + 1).
a(n) = n * A069099(n).

A372184 a(n) = 2^(1-3n)((2*n)!)^2/n.

Original entry on oeis.org

1, 9, 675, 198450, 160744500, 291751267500, 1035425248357500, 6523179064652250000, 67867154988642009000000, 1102501932790489436205000000, 26741184379833321275152275000000, 933641711437500579000666529350000000, 45515033432578153226282493305812500000000

Offset: 1

Thomas Scheuerle, Apr 21 2024

nonn

Cf. A370411, A370412.

Cf. A003658, A139757, A134372, A339483.

PARI
```
a(n) = 2^(1-3*n)*(2*n)!^2/n
```

a(n) = Product_{k=1..n-1} A339483(k).
a(n) = A139757(n-1)*8^(1-n)*A134372(n-1).
a(n) = (2^n*n*2*Zeta_k(1-2*n)*Pi^(4*n))/(D_k^(2*n-1)*sqrt(D_k)*Zeta_k(2*n)) where Zeta_k() is the Dedekind zeta function over a real quadratic field with fundamental discriminant D_k = A003658(m) for some m > 1.
a(n) = 8^(1-n)*Integral_{x>=0} ( x^(2*n-(1/2))*BesselK(1, 2*sqrt(x)) ), where BesselK(m, ...) is the modified Bessel function K_m(...) of the first kind.
Sum_{n>=1} (x^(n-1)/a(n)) = (BesselI(1, 2*2^(3/4)*x^(1/4)) - BesselJ(1, 2*2^(3/4)*x^(1/4)))/(4*2^(1/4)*x^(3/4))
= (d/dx)((-2 + BesselI(0, 2*2^(3/4)*x^(1/4)) + BesselJ(0, 2*2^(3/4)*x^(1/4)))/4), where BesselI(m, ...) is the modified Bessel function I_m(...) of the first kind and BesselJ(m, ...) is the Bessel function J_m(...) of the first kind.
a(n) ~ Pi*2^(n+3)*exp(-4*n)*n^(4*n). - Stefano Spezia, Apr 22 2024
D-finite with recurrence 2*a(n) -n*(n-1)*(-1+2*n)^2*a(n-1)=0. - R. J. Mathar, May 20 2024

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A015223 Odd pentagonal pyramidal numbers.

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A144849 Coefficients in the expansion of the squared sine lemniscate function.

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A329530 a(n) = n * (7*binomial(n, 2) + 1).

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A372184 a(n) = 2^(1-3*n)*((2*n)!)^2/n.

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A372184 a(n) = 2^(1-3n)((2*n)!)^2/n.