Search: a002896 -id:a002896
|
| |
| |
|
|
|
1, 7, 97, 1957, 46687, 1219243, 33715399, 970085119, 28740443449, 870830918389, 26860099935529, 840549807424369, 26620996978712269, 851664885506669269, 27482469263443730269, 893460843597349019629, 29235859228655427097639
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) ~ 2^(2*n) * 3^(2*n + 7/2) / (35 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Feb 17 2024
|
|
|
EXAMPLE
|
a(4) = 1 + 6 + 90 + 1860 + 44730 = 46687.
|
|
|
MATHEMATICA
|
b[n_] := b[n] = (* A002896 *) Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n}, {1, 1}, 4]; a[n_] := Sum[b[k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Dec 20 2011 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A086231
|
|
Decimal expansion of value of Watson's integral.
|
|
+10
14
|
|
|
|
1, 5, 1, 6, 3, 8, 6, 0, 5, 9, 1, 5, 1, 9, 7, 8, 0, 1, 8, 1, 5, 6, 0, 1, 2, 1, 5, 9, 6, 8, 1, 4, 2, 0, 7, 7, 9, 9, 5, 5, 3, 8, 7, 0, 4, 4, 4, 5, 2, 2, 6, 2, 6, 7, 6, 5, 6, 6, 9, 8, 0, 4, 6, 3, 6, 5, 8, 0, 8, 6, 3, 2, 0, 3, 5, 3, 5, 2, 1, 4, 5, 0, 4, 0, 1, 6, 1, 1, 7, 4, 1, 2, 0, 9, 6, 8, 8, 1, 1, 3, 9, 2
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
George N. Watson, Three triple integrals, The Quarterly Journal of Mathematics, Vol. os-10, No. 1 (1939), pp. 266-276.
|
|
|
FORMULA
|
Equals (sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2/(32*Pi^3). - G. C. Greubel, Jan 07 2018
|
|
|
EXAMPLE
|
1.51638605915197801815601215968142077995538704445226267656698...
|
|
|
MAPLE
|
evalf((sqrt(3)-1)*(GAMMA(1/24)*GAMMA(11/24))^2 / (32*Pi^3), 120); # Vaclav Kotesovec, Sep 16 2014
|
|
|
MATHEMATICA
|
|
|
|
PROG
|
(PARI) (sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2 / (32*Pi^3) \\ Altug Alkan, Apr 13 2016
(Magma) C<i> := ComplexField(); (Sqrt(3)-1)*(Gamma(1/24)*Gamma(11/24))^2/(32*Pi(C)^3) // G. C. Greubel, Jan 07 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A084261
|
|
A binomial transform of factorial numbers.
|
|
+10
12
|
|
|
|
1, 1, 2, 4, 9, 21, 52, 134, 361, 1009, 2926, 8768, 27121, 86373, 282864, 950866, 3277169, 11564353, 41739130, 153919324, 579411641, 2224535125, 8703993420, 34681783422, 140637608089, 580019801201, 2431509498406, 10355296410712
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Binomial transform of A000142 (with interpolated zeros).
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*k!.
a(n) = Sum_{k=0..n} C(n, k)*(k/2)!*((1+(-1)^k)/2) .
E.g.f.: exp(x)*(1+sqrt(Pi)/2*x*exp(x^2/4)*erf(x/2)). - Vladeta Jovovic, Sep 25 2003
O.g.f.: A(x) = 1/(1-x-x^2/(1-x-x^2/(1-x-2*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-[(n+1)/2]*x^2/(1- ...))))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
a_n ~ (1/2) * sqrt(Pi*n/e)*(n/2)^(n/2)*exp(-n/2 + sqrt(2n)). - Cecil C Rousseau (ccrousse(AT)memphis.edu), Mar 14 2006: (cf. A002896).
Conjecture: 2*a(n) -4*a(n-1) +(-n+2)*a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Nov 30 2012
|
|
|
MATHEMATICA
|
Table[Sum[Binomial[n, 2*k]*k!, {k, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, Jan 24 2017 *)
|
|
|
PROG
|
(PARI) for(n=0, 50, print1(sum(k=0, floor(n/2), binomial(n, 2*k)*k!), ", ")) \\ G. C. Greubel, Jan 24 2017
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A039699
|
|
Number of 4-dimensional cubic lattice walks that start and end at the origin after 2n steps, free to pass through origin at intermediate stages.
|
|
+10
11
|
|
|
|
1, 8, 168, 5120, 190120, 7939008, 357713664, 16993726464, 839358285480, 42714450658880, 2225741588095168, 118227198981126144, 6380762273973278464, 349019710593278412800, 19310744204362333900800, 1079054103459778710405120, 60818479243449308702049960
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Generating function G(x) is D-finite with a singular point at x = 1/64 (cf. Graph Link). After summing 300000 terms, G(1/64) = 1.239466... and 1 - 1/G(1/64) = 0.193201... Convergence to A086232 is very slow. - Bradley Klee, Aug 20 2018
a(n) is also the constant term in the expansion of (w + 1/w + x + 1/x + y + 1/y + z + 1/z)^(2n). This follows directly from the sequence name, each variable corresponding to a single step in one of the four axis directions. - Christopher J. Smyth, Sep 28 2018
|
|
|
REFERENCES
|
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f.: Sum_{n>=0} a(2*n) * x^(2*n)/(2*n)! = I_0(2*x)^4. (I = Modified Bessel function of the first kind).
a(n) = binomial(2*n,n)^2*hypergeom([1/2,-n,-n,-n],[1,1,1/2-n],1). - Peter Luschny, May 23 2017
G.f.: Define G(x) = Sum_{n>=0} a(n)*x^n and G^(j) = (d/dx)^j G(x), then Sum_{j=0..4,k=0..5} M_{j,k}*G^(j)*x^k = 0, with
M={{-8, 768, 0, 0, 0, 0}, {1, -424, 14592, 0, 0, 0}, {0, 7, -1172, 25344, 0, 0}, {0, 0, 6, -640, 10240, 0}, {0, 0, 0, 1, -80, 1024}}.
Sum_{j=0..2,k=0..4} M_{j,k}*a(n-j)*n^k = 0, with
M={{0, 0, 0, 0, 1}, {-8, 52, -132, 160, -80}, {768, -3584, 5888, -4096, 1024}}.
(End)
a(n) = Sum_{i+j+k+l=n, 0<=i,j,k,l<=n} multinomial(2n [i,i,j,j,k,k,l,l]). - Shel Kaphan, Jan 16 2023
|
|
|
EXAMPLE
|
a(5)=7939008, i.e., there are 7939008 different walks that start and end at origin of a 4-dimensional integer lattice after 2*5=10 steps, free to pass through origin at intermediate steps.
|
|
|
MAPLE
|
A039699 := n -> binomial(2*n, n)^2*hypergeom([1/2, -n, -n, -n], [1, 1, 1/2 - n], 1):
|
|
|
MATHEMATICA
|
max = 30 (* must be even *); Partition[ CoefficientList[ Series[ BesselI[0, 2 x]^4, {x, 0, max}], x]*Range[0, max]!, 2][[All, 1]] (* Jean-François Alcover, Oct 05 2011 *)
With[{nn=30}, Take[CoefficientList[Series[BesselI[0, 2x]^4, {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Aug 09 2013 *)
RecurrenceTable[{256*(n-1)^2*(2*n-3)*(2*n-1)*a[n-2] - 4*(2*n-1)^2*(5*n^2-5*n+2)*a[n-1] + n^4*a[n]==0, a[0]==1, a[1]==8}, a, {n, 0, 100}] (* Bradley Klee, Aug 20 2018 *)
|
|
|
PROG
|
(PARI)
C=binomial;
A002895(n) = sum(k=0, n, C(n, k)^2 * C(2*n-2*k, n-k) * C(2*k, k) );
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,nice,easy,walk
|
|
|
AUTHOR
|
Alessandro Zinani (alzinani(AT)tin.it)
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A049037
|
|
Number of cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.
|
|
+10
9
|
|
|
|
1, 6, 54, 996, 22734, 577692, 15680628, 445162392, 13055851998, 392475442092, 12029082873372, 374482032292008, 11808861461931492, 376406128925067528, 12108063535794336312, 392560994063887113744, 12814685828476778001726, 420836267423433182275404
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
REFERENCES
|
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
|
|
|
LINKS
|
|
|
|
FORMULA
|
Define a_0, a_1, ... = [ 1, 6, 54, ... ] by 1+Sum b_i x^i = 1/(1-Sum a_i x^i) where b_0, b_1, ... = [ 1, 6, 90, ... ] = A002896.
Or, Sum[ a(n) x^(2n), n=1, 2, ...infinity ] = 1-1/Sum[ A002896(n)*x^(2n), n=0, 1, ...infinity ].
G.f.: 2-sqrt(1+12*z) /hypergeom([1/8, 3/8], [1], 64/81*z *(1+sqrt(1-36*z))^2 *(2+sqrt(1-36*z))^4 /(1+12*z)^4)/ hypergeom([1/8, 3/8], [1], 64/81*z *(1-sqrt(1-36*z))^2 *(2-sqrt(1-36*z))^4 /(1+12*z)^4). - Sergey Perepechko, Jan 30 2011
a(n) ~ c * 36^n / n^(3/2), where c = 0.1014559485279103938501072426734... . - Vaclav Kotesovec, Sep 13 2014
c = 384 * (3 + 2*sqrt(3)) * Pi^(9/2) / (Gamma(1/24)^4 * Gamma(11/24)^4). - Vaclav Kotesovec, Apr 23 2023
|
|
|
EXAMPLE
|
a(5) = 577692 because there are 577692 different walks that start and end at the origin after 2*5=10 steps, avoiding origin at intermediate steps.
|
|
|
MAPLE
|
read transforms; t1 := [ seq(A002896(i), i=1..25) ]; INVERTi(t1);
# second Maple program:
b:= proc(n) option remember; `if`(n<2, 5*n+1,
(2*(2*n-1)*(10*n^2-10*n+3) *b(n-1)
-36*(n-1)*(2*n-1)*(2*n-3) *b(n-2)) /n^3)
end:
g:= proc(n) g(n):= `if` (n<1, -1, -add(g(n-i) *b(i), i=1..n)) end:
a:= n-> abs(g(n)):
|
|
|
MATHEMATICA
|
(* A002896 : *) b[n_] := b[n] = Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n}, {1, 1}, 4]; max = 32; a[0] = 1; se = Series[ Sum[ a[n] x^(2 n), {n, 1, max}] - 1 + 1/Sum[ b[n]*x^(2 n), {n, 0, max}], {x, 0, max}]; coes = CoefficientList[se, x]; sol = First[ Solve[ Thread[ coes == 0]]]; Table[ a[n], {n, 0, 16}] /. sol (* Jean-François Alcover, Dec 20 2011 *)
b[n_] := b[n] = If[n < 2, 5*n + 1, (2*(2*n - 1)*(10*n^2 - 10*n + 3)*b[n-1] - 36*(n - 1)*(2*n - 1)*(2*n - 3)*b[n-2]) / n^3];
g[n_] := g[n] = If[n < 1, -1, -Sum [g[n - i]*b[i], {i, 1, n}]];
a[n_] := Abs[g[n]];
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
easy,nonn,nice
|
|
|
AUTHOR
|
Alessandro Zinani (alzinani(AT)tin.it)
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A287318
|
|
Square array A(n,k) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.
|
|
+10
8
|
|
|
|
1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 36, 20, 0, 1, 8, 90, 400, 70, 0, 1, 10, 168, 1860, 4900, 252, 0, 1, 12, 270, 5120, 44730, 63504, 924, 0, 1, 14, 396, 10900, 190120, 1172556, 853776, 3432, 0, 1, 16, 546, 19920, 551950, 7939008, 32496156, 11778624, 12870, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
LINKS
|
|
|
|
FORMULA
|
A(n,k) = A287316(n,k) * binomial(2*n,n).
|
|
|
EXAMPLE
|
Arrays start:
k\n| 0 1 2 3 4 5 6
---|---------------------------------------------------------
k=0| 1, 0, 0, 0, 0, 0, 0, ... A000007
k=1| 1, 2, 6, 20, 70, 252, 924, ... A000984
k=2| 1, 4, 36, 400, 4900, 63504, 853776, ... A002894
k=3| 1, 6, 90, 1860, 44730, 1172556, 32496156, ... A002896
k=4| 1, 8, 168, 5120, 190120, 7939008, 357713664, ... A039699
k=5| 1, 10, 270, 10900, 551950, 32232060, 2070891900, ... A287317
k=6| 1, 12, 396, 19920, 1281420, 96807312, 8175770064, ... A356258
k=7| 1, 14, 546, 32900, 2570050, 238935564, 25142196156, ...
k=8| 1, 16, 720, 50560, 4649680, 514031616, 64941883776, ...
k=9| 1, 18, 918, 73620, 7792470, 999283068, 147563170524, ...
|
|
|
MAPLE
|
A287318_row := proc(k, len) local b, ser;
b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
seq((2*i)!*coeff(ser, x, i), i=0..len-1) end:
for k from 0 to 6 do A287318_row(k, 9) od;
|
|
|
MATHEMATICA
|
Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (2 n)!, {n, 0, 6}], {k, 0, 6}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A288458
|
|
Chebyshev coefficients of density of states of cubic lattice.
|
|
+10
8
|
|
|
|
1, -24, 288, -2688, -32256, 2820096, -95035392, 1972076544, -9841803264, -1288894414848, 70351960670208, -2164060518875136, 36664809432809472, 365875642245316608, -55960058736918134784, 2436570173137823465472, -64272155689216515244032, 664295705652718630600704, 35692460661517822602510336
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
This is the sequence of integers z^n g_n for n=0,2,4,6,... where g_n are the coefficients in the Chebyshev polynomial expansion of the density of states of the simple cubic lattice (z=6), g(w) = 1 / (Pi*sqrt(1-w^2)) * Sum_{n>=0} (2-delta_n) g_n T_n(w). Here |w| <= 1 and delta is the Kronecker delta.
The Chebyshev coefficients, g_n, are related to the number of walks on the lattice that return to the origin, W_n, as g_n = Sum_{k=0..n} a_{nk} z^{-k} W_k, where z is the coordination number of the lattice and a_{nk} are the coefficients of Chebyshev polynomials such that T_n(x) = Sum_{k=0..n} a_{nk} x^k.
The author was unable to obtain a closed form for z^n g_n.
|
|
|
LINKS
|
|
|
|
MATHEMATICA
|
Whon[n_] := If[OddQ[n], 0,
Sum[Binomial[n/2, j]^2 Binomial[2j, j], {j, 0, n/2}]];
Wcub[n_] := Binomial[n, n/2] Whon[n];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*6^(n-k)*Wcub[k], {k, 0, n}];
Table[zng[n], {n, 0, 50}]
|
|
|
CROSSREFS
|
Related to numbers of walks returning to origin, W_n, on cubic lattice (A002896).
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A001413
|
|
Number of 2n-step polygons on cubic lattice.
(Formerly M5154 N2238)
|
|
+10
7
|
|
|
|
0, 24, 264, 3312, 48240, 762096, 12673920, 218904768, 3891176352, 70742410800, 1309643747808, 24609869536800, 468270744898944, 9005391024862848, 174776445357365040, 3419171337633496704
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
a(n) is the number of 2n-step closed self-avoiding paths on the cubic lattice. - Bert Dobbelaere, Jan 04 2019
|
|
|
REFERENCES
|
B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 462.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
CROSSREFS
|
Cf. A010566 (for square lattice equivalent).
Cf. A002896 (without self-avoidance restriction).
|
|
|
KEYWORD
|
nonn,walk,more
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A287317
|
|
Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.
|
|
+10
6
|
|
|
|
1, 10, 270, 10900, 551950, 32232060, 2070891900, 142317232200, 10277494548750, 770878551371500, 59577647564312020, 4717432065143561400, 381091087190569291900, 31308955091335405435000, 2609450031306515140215000, 220199552765301571338488400
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^5.
a(n) ~ 2^(2*n) * 5^(2*n + 5/2) / (16 * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Nov 13 2017
a(n) = Sum_{i+j+k+l+m=n, 0<=i,j,k,l,m<=n} multinomial(2n, [i,i,j,j,k,k,l,l,m,m]). - Shel Kaphan, Jan 24 2023
|
|
|
MAPLE
|
A287317_list := proc(len) series(BesselI(0, 2*sqrt(x))^5, x, len);
seq((2*i)!*coeff(%, x, i), i=0..len-1) end: A287317_list(16);
|
|
|
MATHEMATICA
|
Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^5, {x, 0, n}] (2 n) !, {n, 0, 15}]
Table[Binomial[2n, n]^2 Sum[(Binomial[n, j]^4/Binomial[2n, 2j]) HypergeometricPFQ[{-j, -j, -j}, {1, 1/2-j}, 1/4], {j, 0, n}], {n, 0, 15}]
Table[Sum[(2 n)!/(i! j! k! l! (n-i-j-k-l)!)^2, {i, 0, n}, {j, 0, n-i}, {k, 0, n-i-j}, {l, 0, n-i-j-k}], {n, 0, 30}] (* Shel Kaphan, Jan 24 2023 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy,walk
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Moved original definition to formula section and reworded definition descriptively similar to sequence A039699, by Dave R.M. Langers, Oct 12 2022
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A303503
|
|
a(n) = (2*n)! * [x^(2*n)] BesselI(0,2*x)^n.
|
|
+10
3
|
|
|
|
1, 2, 36, 1860, 190120, 32232060, 8175770064, 2898980908824, 1369263687414480, 830988068906518380, 630109741730668410640, 583773362067938664133512, 648851848280206013365243776, 852146184628067383511375555000, 1305460597778526044143501996708800, 2307324514460203126471248458864413200
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) ~ c * d^n * n^(2*n), where c = 1.72802011936236389522137050964080... and d = 1.1381284656425793765251319541847869000364101065484286935... - Vaclav Kotesovec, Apr 26 2018
|
|
|
MAPLE
|
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
end:
a:= n-> (2*n)!*b(n$2)/n!^2:
|
|
|
MATHEMATICA
|
Table[(2 n)! SeriesCoefficient[BesselI[0, 2 x]^n, {x, 0, 2 n}], {n, 0, 15}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.020 seconds
|
