A057703 -id:A057703 - cp's OEIS frontend

A036917 G.f.: (4/Pi^2)EllipticK(4x^(1/2))^2.

1, 8, 88, 1088, 14296, 195008, 2728384, 38879744, 561787864, 8206324928, 120929313088, 1794924383744, 26802975999424, 402298219288064, 6064992788397568, 91786654611673088, 1393772628452578264, 21227503080738294464, 324160111169327247424

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 8*x + 88*x^2 +  1088*x^3 + 14296*x^5 + 195008*x^5 + ... - _Michael Somos_, May 29 2023

M. Petkovsek et al., "A=B", Peters, p. ix of second printing.

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
B. Adamczewski, J. P. Bell, and E. Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Ji-Cai Liu and He-Xia Ni, Supercongruences for Almkvist--Zudilin sequences, arXiv:2004.07652 [math.NT], 2020. See Vn.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020. See Vn.
Zhi-Hong Sun, Congruences for two types of Apery-like sequences, arXiv:2005.02081 [math.NT], 2020.

Cf. A002894, A036915, A057703.
Cf. A007318, A036916, A036829.
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a036917 n = sum $ map
   (\k -> (a007318 (2*n-2*k) (n-k))^2 * (a007318 (2*k) k)^2) [0..n]
-- Reinhard Zumkeller, May 24 2012

a[n_] := (16 (n - 1/2)(2*n^2 - 2*n + 1)a[n - 1] - 256(n - 1)^3 a[n - 2])/n^3; a[0] = 1; a[1] = 8; Array[a, 19, 0] (* Or *)
f[n_] := Sum[(Binomial[2 (n - k), n - k] Binomial[2 k, k])^2, {k, 0, n}]; Array[f, 19, 0] (* Or *)
lmt = 20; Take[ 4^Range[0, 2 lmt]*CoefficientList[ Series[(4/Pi^2) EllipticK[4 x^(1/2)]^2, {x, 0, lmt}], x^(1/2)], lmt] (* Robert G. Wilson v *)
a[n_] := HypergeometricPFQ[{1/2, 1/2, -n, -n}, {1, 1/2-n, 1/2-n}, 1] * 4^n * (2n-1)!!^2 / n!^2 (* Vladimir Reshetnikov, Mar 08 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, EllipticNomeQ[16*x]]^4, {x, 0, n}]; (* Michael Somos, May 30 2023 *)

for(n=0,25, print1(sum(k=0,n, (binomial(2*n-2*k,n-k) *binomial(2*k,k))^2), ", ")) \\ G. C. Greubel, Oct 24 2017

a(n) = if(n<0, 0, polcoeff(agm(1, sqrt(1 - 16*x + x*O(x^n)))^-2, n)); /* Michael Somos, May 29 2023 */

a(n) = (16*(n-1/2)*(2*n^2-2*n+1)*a(n-1)-256*(n-1)^3*a(n-2))/n^3.
a(n) = Sum_{k=0..n} (C(2 * (n-k), n-k) * C(2 * k, k))^2. [corrected by Tito Piezas III, Oct 19 2010]
a(n) = hypergeom([1/2, 1/2, -n, -n], [1, 1/2-n, 1/2-n], 1) * 4^n * (2n-1)!!^2 / n!^2. - Vladimir Reshetnikov, Mar 08 2014
a(n) ~ 2^(4*n+1) * log(n) / (n*Pi^2) * (1 + (4*log(2) + gamma)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Nov 28 2015
G.f. y=A(x) satisfies: 0 = x^2*(16*x - 1)^2*y''' + 3*x*(16*x - 1)*(32*x - 1)*y'' + (1792*x^2 - 112*x + 1)*y' + 8*(32*x - 1)*y. - Gheorghe Coserea, Jul 03 2018
G.f.: 1 / AGM(1, sqrt(1 - 16*x))^2. - Vaclav Kotesovec, Oct 01 2019
It appears that a(n) is equal to the coefficient of (x*y*z*t)^n in the expansion of (1+x+y+z-t)^n * (1+x+y-z+t)^n * (1+x-y+z+t)^n * (1-x+y+z+t)^n. Cf. A000172. - Peter Bala, Sep 21 2021
G.f. y = A(x) satisfies 0 = x*(1 - 16*x)*(2*y''*y - y'*y') + 2*(1 - 32*x)*y*y' - 16*y*y. - Michael Somos, May 29 2023
Expansion of theta_3(0, q)^4 in powers of m/16 where the modulus m = k^2. - Michael Somos, May 30 2023
From Paul D. Hanna, Mar 25 2024: (Start)
G.f. ( Sum_{n>=0} binomial(2*n,n)^2 * x^n )^2.
G.f. Sum_{n>=0} binomial(2*n,n)^3 * x^n * (1 - 16*x)^n. (End)

Replaced complicated definition via a formula with simple generating function provided by Vladeta Jovovic, Dec 01 2003. Thanks to Paul D. Hanna for suggesting this. - N. J. A. Sloane, Mar 25 2024

A006261 a(n) = Sum_{k=0..5} binomial(n,k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 63, 120, 219, 382, 638, 1024, 1586, 2380, 3473, 4944, 6885, 9402, 12616, 16664, 21700, 27896, 35443, 44552, 55455, 68406, 83682, 101584, 122438, 146596, 174437, 206368, 242825, 284274, 331212, 384168, 443704, 510416, 584935, 667928, 760099, 862190

Offset: 0

N. J. A. Sloane, based on a suggestion from S. C. Chan, Jun 10 1975

a(n) is the sum of the first six terms of the n-th row in Pascal's triangle. - Geoffrey Critzer, Jan 19 2009
Also the interpolating polynomial for the divisors of 32: {a(k): 0 <= k < 6} = {1,2,4,8,16,32}. - Reinhard Zumkeller, Jun 17 2009
a(n) is the maximal number of regions in 5-space formed by n-1 4-dimensional hypercubes. - Carl Schildkraut, May 26 2015
a(n) is the number of binary words of length n matching the regular expression 1*0*1*0*1*0*. A000124, A000125, A000127 count binary words of the form 0*1*0*, 1*0*1*0*, and 0*1*0*1*0*, respectively. - Manfred Scheucher, Jun 22 2023

			a(7) = 120 because the first six terms in the 7th row of Pascal's triangle 1 + 7 + 21 + 35 + 35 + 21 = 120. - _Geoffrey Critzer_, Jan 19 2009

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. C. Chan, Letter to N. J. A. Sloane, Oct. 1975
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32.
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)
R. K. Guy, Letter to N. J. A. Sloane
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A057703.

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A007318, A008859, A008860, A008861, A008862, A008863, A219531.

a006261 = sum . take 6 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

[(n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

A006261:=(z**2-z+1)*(3*z**2-3*z+1)/(z-1)**6; # Simon Plouffe in his 1992 dissertation

CoefficientList[
  Series[(1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120) Exp[x], {x, 0,
    52}], x]*Table[n!, {n, 0, 52}]

a(n)=sum(k=0,5,binomial(n,k)) \\ Charles R Greathouse IV, Apr 08 2016

A006261_list, m = [], [1, -3, 4, -2, 1, 1]
for _ in range(10**2):
    A006261_list.append(m[-1])
    for i in range(5):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

[binomial(n,1)+binomial(n,3)+binomial(n,5) for n in range(1, 38)] # Zerinvary Lajos, May 17 2009

a(n) = A057703(n) + 1.
a(n) = binomial(n+1, 5) + binomial(n+1, 3) + binomial(n+1, 1). - Len Smiley, Oct 20 2001
G.f.: (1 - 4*x + 7*x^2 - 6*x^3 + 3*x^4)/(1-x)^6. - Geoffrey Critzer, Jan 19 2009
E.g.f.: (1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120)*exp(x).
a(n) = (n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120. - Reinhard Zumkeller, Jun 17 2009
a(n) = a(n-1) + A000127(n-1). - Christian Schroeder, Jan 04 2016

A115567 a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 63, 126, 246, 465, 847, 1485, 2509, 4095, 6475, 9948, 14892, 21777, 31179, 43795, 60459, 82159, 110055, 145498, 190050, 245505, 313911, 397593, 499177, 621615, 768211, 942648, 1149016, 1391841, 1676115, 2007327, 2391495

Offset: 0

Jonathan Vos Post, Mar 12 2006

a(n) = n + T(n) + Tet(n) + Ptop(n) + 5-Simplex(n) + 6-Simplex(n), where T(n) = n-th triangular number A000217(n), Tet(n) = n-th tetrahedral number A000292(n), Ptop(n) = n-th pentatope number A000332(n), 5-Simplex(n) = n-th 5-simplex number A000389(n), 6-Simplex(n) = n-th 6-simplex number A000579(n).

By analogy to A004006, A055795 and A057703, I presume that a(n) = Answer to the question: if you have a tall building and 6 plates and you need to find the highest story, a plate thrown from which does not break, what is the number of stories you can handle given n tries?

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. [_Parthasarathy Nambi_, Sep 30 2009]
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000217, A000332, A000389, A000292, A000579, A004006, A055795, A057703.

[n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720: n in [0..30]]; // G. C. Greubel, Nov 25 2017

seq(sum(binomial(n,k),k=1..6),n=0..36); # Zerinvary Lajos, Dec 13 2007

Table[n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720, {n,0,30}] (* G. C. Greubel, Nov 25 2017 *)

for(n=0,30, print1(n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720, ", ")) \\ G. C. Greubel, Nov 25 2017

[binomial(n,2)+binomial(n,4)+binomial(n,6) for n in range(1, 38)] # Zerinvary Lajos, May 17 2009

[binomial(n,1)+binomial(n,3)+binomial(n,5)+binomial(n,2)+binomial(n,4)+binomial(n,6) for n in range(0, 37)] # Zerinvary Lajos, May 17 2009

a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).
a(n) = A000579(n) + A000389(n) + A000332(n) + A000292(n) + A000217(n) + n.
a(n) = A000579(n) + A057703(n).
G.f.: x*(1-x+x^2)*(1-3*x+3*x^2)/(1-x)^7. - Colin Barker, Mar 16 2012
From G. C. Greubel, Nov 25 2017: (Start)
a(n) = n*(n + 1)*(n^4 - 10*n^3 + 65*n^2 - 140*n + 444)/720.
E.g.f.: x*(720 + 360*x + 120*x^2 + 30*x^3 + 6*x^4 + x^5)*exp(x)/720. (End)

A116082 a(n) = C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 63, 127, 254, 501, 967, 1815, 3301, 5811, 9907, 16383, 26332, 41225, 63003, 94183, 137979, 198439, 280599, 390655, 536154, 726205, 971711, 1285623, 1683217, 2182395, 2804011, 3572223, 4514872, 5663889, 7055731, 8731847

Offset: 0

Jonathan Vos Post, Mar 13 2006

Number of compositions with at most three parts distinct from 1 and with a sum at most n. - Beimar Naranjo, Mar 12 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. See Table 2 on page 370. [_Parthasarathy Nambi_, Jun 18 2009]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000217, A000225, A004006, A055795, A057703, A115567, A116082. [Parthasarathy Nambi, Jun 18 2009]

[n*(n^6-14*n^5+112*n^4-350*n^3+1099*n^2+364*n+3828)/5040: n in [0..40]]; // Vincenzo Librandi, Jun 21 2011

a:=n->n*(n^6-14*n^5+112*n^4-350*n^3+1099*n^2+364*n+3828)/5040: seq(a(n),n=0..35); # Emeric Deutsch, Apr 14 2006
seq(sum(binomial(n,k),k=1..7),n=0..35); # Zerinvary Lajos, Dec 14 2007

Table[Total[Binomial[n,Range[7]]],{n,0,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{0,1,3,7,15,31,63,127},41](* Harvey P. Dale, Aug 05 2011 *)

for(n=0,30, print1(n*(n^6 -14*n^5 +112*n^4 -350*n^3 +1099*n^2 +364*n +3828)/5040, ", ")) \\ G. C. Greubel, Nov 25 2017

a(n) = A000580(n) + A000579(n) + A000389(n) + A000332(n) + A000292(n) + A000217(n) + n.
a(n) = A000580(n) + A115567(n).
a(n) = n*(n^6 - 14*n^5 + 112*n^4 - 350*n^3 + 1099*n^2 + 364*n + 3828)/5040. - Emeric Deutsch, Apr 14 2006
G.f.: x*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)/(1-x)^8. - R. J. Mathar, Jun 20 2011
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), with a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(4)=15, a(5)=31, a(6)=63, a(7)=127. - Harvey P. Dale, Aug 05 2011

A131251 A000012 * A052509.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 3, 1, 5, 10, 7, 3, 1, 6, 15, 14, 7, 3, 1, 7, 21, 25, 15, 7, 3, 1, 8, 28, 41, 30, 15, 7, 3, 1, 9, 36, 63, 56, 31, 15, 7, 3, 1, 10, 45, 92, 98, 62, 31, 15, 7, 3, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A001924: (1, 3, 7, 14, 26, 46, 79, ...). A131252 = A052509 * A000012.
From Clark Kimberling, Feb 07 2011: (Start)
When formatted as a rectangle R with northwest corner
  1, 2, 3,  4,  5,  6, ...
  1, 3, 6, 10, 15, 21, ...
  1, 3, 7, 14, 25, 41, ...
  1, 3, 7, 15, 30, 56, ...
  1, 3, 7, 15, 31, 62, ...
  ...
the following properties hold:
R is the accumulation array of the transpose of A052553 (a version of Pascal's triangle); see A144112 for the definition of accumulation array.
row 1: A000027
row 2: A000217
row 3: A004006
row 4: A055795
row 5: A057703
row 6: A115567
limiting row:  A000225
antidiagonal sums: A001924.
(End)

			First few rows of the triangle:
  1;
  2,  1;
  3,  3,  1;
  4,  6,  3,  1;
  5, 10,  7,  3,  1;
  6, 15, 14,  7,  3,  1;
  7, 21, 25, 15,  7,  3,  1;
  ...

Cf. A052509, A000012, A001924, A131252.

A000012 * A052509 as infinite lower triangular matrices.

cp's OEIS Frontend

A036917 G.f.: (4/Pi^2)*EllipticK(4*x^(1/2))^2.

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A006261 a(n) = Sum_{k=0..5} binomial(n,k).

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A115567 a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

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A116082 a(n) = C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

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A131251 A000012 * A052509.

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A036917 G.f.: (4/Pi^2)EllipticK(4x^(1/2))^2.