A000302 -id:A000302 - cp's OEIS frontend

A038845 3-fold convolution of A000302 (powers of 4).

1, 12, 96, 640, 3840, 21504, 114688, 589824, 2949120, 14417920, 69206016, 327155712, 1526726656, 7046430720, 32212254720, 146028888064, 657129996288, 2937757630464, 13056700579840, 57724360458240, 253987186016256, 1112705767309312, 4855443348258816

Offset: 0

Wolfdieter Lang

Also convolution of A002802 with A000984 (central binomial coefficients).
With a different offset, number of n-permutations of 5 objects u, v, w, z, x with repetition allowed, containing exactly two u's. - Zerinvary Lajos, Dec 29 2007
Also convolution of A000302 with A002697, also convolution of A002457 with itself. - Rui Duarte, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Adam Ehrenberg, Joseph T. Iosue, Abhinav Deshpande, Dominik Hangleiter, and Alexey V. Gorshkov, The Second Moment of Hafnians in Gaussian Boson Sampling, arXiv:2403.13878 [quant-ph], 2024. See p. 30.
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A000302, A000984, A002457, A002697, A002802, A038231, A052780.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), this sequence (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

List([0..30], n-> 4^n*Binomial(n+2,n) ); # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+2, 2): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq((n+2)*(n+1)*4^n/2, n=0..30); # Zerinvary Lajos, Apr 25 2007

Table[4^n*Binomial[n+2,n], {n,0,30}] (* G. C. Greubel, Jul 20 2019 *)

a(n)=(n+2)*(n+1)<<(2*n-1) \\ Charles R Greathouse IV, Aug 21 2015

[4^(n-2)*binomial(n,2) for n in range(2, 30)] # Zerinvary Lajos, Mar 11 2009

a(n) = (n+2)*(n+1)*2^(2*n-1).
G.f.: 1/(1-4*x)^3.
a(n) = Sum_{u+v+w+x+y+z=n} f(u)*f(v)*f(w)*f(x)*f(y)*f(z) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004
a(n) = binomial(n+2,n) * 4^n. - Rui Duarte, Oct 08 2011
E.g.f.: (1 + 8*x + 8*x^2)*exp(4*x). - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 8 - 24*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 40*log(5/4) - 8. (End)

A038846 4-fold convolution of A000302 (powers of 4); expansion of g.f. 1/(1-4*x)^4.

Original entry on oeis.org

1, 16, 160, 1280, 8960, 57344, 344064, 1966080, 10813440, 57671680, 299892736, 1526726656, 7633633280, 37580963840, 182536110080, 876173328384, 4161823309824, 19585050869760, 91396904058880, 423311976693760, 1947235092791296, 8901646138474496, 40462027902156800

Offset: 0

Wolfdieter Lang

Also minimal 3-covers of a labeled n-set that cover 3 points of that set uniquely (if offset is 3). Cf. A057524 for unlabeled case. - Vladeta Jovovic, Sep 02 2000
Also convolution of A020918 with A000984 (central binomial coefficients).
Let M=[1,0,0,i;0,1,i,0;0,i,1,0;i,0,0,1], i=sqrt(-1). Then 1/det(I-xM) = 1/(1-4x)^4. - Paul Barry, Apr 27 2005
With a different offset, number of n-permutations (n=4) of 5 objects u, v, w, z, x with repetition allowed, containing exactly three u's. Example: a(1)=16 because we have uuuv, uuvu, uvuu, vuuu, uuuw, uuwu, uwuu, wuuu, uuuz, uuzu, uzuu, zuuu, uuux, uuxu, uxuu and xuuu. - Zerinvary Lajos, May 19 2008
From A152818. a(n) = A006044/6. - Paul Curtz, Jan 07 2009
Also convolution of A000302 with A038845, also convolution of A002457 with A002802, also convolution of A002697. - Rui Duarte, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Cf. A000302, A000984, A002457, A002697, A002802, A006044, A020918, A038231, A038845, A057524, A152818.

List([0..30], n-> 4^n*Binomial(n+3,3) ) # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+3, 3): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(seq(binomial(i, j)*4^(i-3), j =i-3), i=3..33); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+3,3)*4^n,n=0..30); # Zerinvary Lajos, May 19 2008

Table[4^n*Binomial[n+3,3], {n,0,30}] (* G. C. Greubel, Jul 20 2019 *)

Vec(1/(1-4*x)^4+O(x^30)) \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number2(n, 4, 0)*binomial(n,3)/2^6 for n in range(3, 33)] # Zerinvary Lajos, Mar 11 2009

a(n) = binomial(n+3, 3)*4^n.
G.f.: 1/(1-4*x)^4.
a(n) = Sum_{a+b+c+d+e+f+g+h=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f)*f(g)*f(h) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 108*log(4/3) - 30.
Sum_{n>=0} (-1)^n/a(n) = 300*log(5/4) - 66. (End)
E.g.f.: exp(4*x)*(3 + 36*x + 72*x^2 + 32*x^3)/3. - Stefano Spezia, Jan 01 2023

A040075 5-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^5.

Original entry on oeis.org

1, 20, 240, 2240, 17920, 129024, 860160, 5406720, 32440320, 187432960, 1049624576, 5725224960, 30534533120, 159719096320, 821412495360, 4161823309824, 20809116549120, 102821517066240, 502682972323840, 2434043865989120, 11683410556747776, 55635288365465600

Offset: 0

Wolfdieter Lang

Also convolution of A020920 with A000984 (central binomial coefficients).
With a different offset, number of n-permutations (n=5) of 5 objects u, v, w, z, x with repetition allowed, containing exactly four (4)u's. Example: a(1)=20 because we have uuuuv, uuuvu, uuvuu, uvuuu, vuuuu, uuuuw, uuuwu, uuwuu, uwuuu, wuuuu, uuuuz, uuuzu, uuzuu, uzuuu, zuuuu, uuuux, uuuxu, uuxuu, uxuuu and xuuuu. - Zerinvary Lajos, May 19 2008
Also convolution of A000302 with A038846, also convolution of A002457 with A020918, also convolution of A002697 with A038845, also convolution of A002802 with A002802. [Rui Duarte, Oct 08 2011]

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Idempotent Number.
Index entries for linear recurrences with constant coefficients, signature (20, -160, 640, -1280, 1024).

Cf. A000302, A020920, A000984.

Cf. A038231.

List([0..30], n-> 4^n*Binomial(n+4, 4)); # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+4, 4): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(seq(binomial(i, j)*4^(i-4), j =i-4), i=4..22); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+4,4)*4^n,n=0..30); # Zerinvary Lajos, May 19 2008
spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, Z, Z, Z, B, B, B, B)}, labeled]: seq(combstruct[count](spec, size=n)/24, n=4..34); # Zerinvary Lajos, Apr 05 2009

Table[Binomial[n+4,4]*4^n, {n,0,30}] (* Michael De Vlieger, Aug 21 2015 *)

vector(30, n, n--; 4^n*binomial(n+4, 4)) \\ G. C. Greubel, Jul 20 2019

[lucas_number2(n, 4, 0)*binomial(n,4)/2^8 for n in range(4, 34)] # Zerinvary Lajos, Mar 11 2009

a(n) = binomial(n+4, 4)*4^n.
G.f.: 1/(1-4*x)^5.
a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10 = n } f(i_1)*f(i_2) *f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) with f(k)=A000984(k). - Rui Duarte, Oct 08 2011
E.g.f.: (3 + 48*x + 144*x^2 + 128*x^3 + 32*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 376/3 - 432*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 2000*log(5/4) - 1336/3. (End)

A163912 Least common multiple of all cycle sizes in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

Original entry on oeis.org

1, 2, 6, 24, 36, 288, 432, 1728, 2592, 31104, 15552

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

See also A163892, A163910, A163911, A163904, A163890.

A163910 Number of cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

Original entry on oeis.org

1, 2, 3, 18, 30, 178, 306, 1864, 3214, 20032, 34708

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

See also A163911, A163912, A163914, A163904, A163890.

A163911 Maximum cycle size in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

Original entry on oeis.org

1, 2, 6, 8, 18, 32, 108, 216, 324, 1944, 1944

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

See also A163891, A163896, A163910, A163912, A163904, A163890.

A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

Original entry on oeis.org

1, 24, 336, 3584, 32256, 258048, 1892352, 12976128, 84344832, 524812288, 3148873728, 18320719872, 103817412608, 574988746752, 3121367482368, 16647293239296, 87398289506304, 452414675091456, 2312341672689664, 11683410556747776, 58417052783738880, 289303499500421120

Offset: 0

Wolfdieter Lang

Also convolution of A020922 with A000984 (central binomial coefficients); also convolution of A040075 with A000302 (powers of 4).
With a different offset, number of n-permutations of 5 objects: u,v,z,x, y with repetition allowed, containing exactly five (5) u's. Example: a(1)=24 because we have uuuuuv uuuuvu uuuvuu uuvuuu uvuuuu vuuuuu uuuuuz uuuuzu uuuzuu uuzuuu uzuuuu zuuuuu uuuuux uuuuxu uuuxuu uuxuuu uxuuuu xuuuuu uuuuuy uuuuyu uuuyuu uuyuuu uyuuuu yuuuuu. - Zerinvary Lajos, Jun 16 2008
Also convolution of A002457 with A020920, also convolution of A002697 with A038846, also convolution of A002802 with A020918, also convolution of A038845 with A038845. - Rui Duarte, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-240,1280,-3840,6144,-4096).

Cf. A038231.

List([0..30], n-> 4^n*Binomial(n+5,5)); # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+5, 5): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(seq(binomial(i+5, j)*4^i, j =i), i=0..30); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+5,5)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008

CoefficientList[Series[1/(1-4x)^6,{x,0,30}],x] (* or *) LinearRecurrence[ {24,-240,1280,-3840,6144,-4096}, {1,24,336,3584,32256, 258048}, 30] (* Harvey P. Dale, Mar 24 2018 *)

Vec(1/(1-4*x)^6 + O(x^30)) \\ Michel Marcus, Aug 21 2015

[lucas_number2(n, 4, 0)*binomial(n,5)/2^10 for n in range(5, 35)] # Zerinvary Lajos, Mar 11 2009

a(n) = binomial(n+5, 5)*4^n.
G.f.: 1/(1-4*x)^6.
a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10+i_11+i_12 = n} f(i_1)* f(i_2)*f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) *f(i_11)*f(i_12), with f(k)=A000984(k). - Rui Duarte, Oct 08 2011
E.g.f.: (15 + 120*x + 240*x^2 + 160*x^3 + 32*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 1620*log(4/3) - 465.
Sum_{n>=0} (-1)^n/a(n) = 12500*log(5/4) - 8365/3. (End)

A163914 Number of 3-cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

Original entry on oeis.org

0, 0, 2, 1, 10, 9, 54, 57, 295, 329, 1613, 1834, 8812, 10072

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

a(n) = A163913(n)/3. Bisections: A163909, A163919. See also A163903, A163911, A163912, A163904, A163890.

A038806 Convolution of A008549 with A000302 (powers of 4).

Original entry on oeis.org

0, 1, 10, 69, 406, 2186, 11124, 54445, 259006, 1205790, 5519020, 24918306, 111250140, 492051124, 2159081192, 9409526397, 40766269774, 175707380630, 753876367356, 3221460111958, 13716223138388, 58210889582796

Offset: 0

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining convex permutominoes, J. Int. Seq. 10 (2007) # 07.9.7

Cf. A000108, A000346, A008549, A000302, A002697.

[(n+3)*4^n -(n+2)*Binomial(2*n+3, n+1)/2: n in [0..25]]; // Vincenzo Librandi, Jun 09 2011

CoefficientList[Series[x ((1 - Sqrt[1 - 4 x])/(2 x)/(1 - 4 x))^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 29 2014 *)

a(n) = (n+3)*4^n -(n+2)*binomial(2*n+3, n+1)/2.
G.f.: x*(c(x)/(1-4*x))^2, where c(x) = g.f. for Catalan numbers A000108.
a(n+1), n >= 0 is convolution of A000346 with itself; a(n+1), n >= 0 is convolution of Catalan numbers A000108 C(n+1), n >= 0 with A002697; a(-1)=0.
Asymptotics: a(n) ~ 4^n*(n+1-4*sqrt(n/Pi)). - Fung Lam, Mar 28 2014
Recurrence: (n-1)*(n+1)*a(n) = 2*(n+1)*(4*n-3)*a(n-1) - 8*n*(2*n+1)*a(n-2). - Vaclav Kotesovec, Mar 28 2014

A054337 7-fold convolution of A000302 (powers of 4).

Original entry on oeis.org

1, 28, 448, 5376, 53760, 473088, 3784704, 28114944, 196804608, 1312030720, 8396996608, 51908706304, 311452237824, 1820797698048, 10404558274560, 58265526337536, 320460394856448, 1734256254517248, 9249366690758656, 48680877319782400, 253140562062868480

Offset: 0

Wolfdieter Lang, Mar 13 2000

With a different offset, number of n-permutations (n>=6) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly six (6) u's. Example: a(1)=28 because we have uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu, vuuuuuu, uuuuuuz, uuuuuzu, uuuuzuu, uuuzuuu, uuzuuuu, uzuuuuu, zuuuuuu, uuuuuux, uuuuuxu, uuuuxuu, uuuxuuu, uuxuuuu, uxuuuuu, xuuuuuu, uuuuuuy, uuuuuyu, uuuuyuu, uuuyuuu, uuyuuuu, uyuuuuu, yuuuuuu. - Zerinvary Lajos, Jun 16 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A000302, A054335.

Cf. A038231.

List([0..30], n-> 4^n*Binomial(n+6,6)); # G. C. Greubel, Jul 21 2019

[4^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(seq(binomial(i, j)*4^(i-6), j =i-6), i=6..36); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+6,6)*4^n,n=0..30); # Zerinvary Lajos, Jun 16 2008

Table[4^n*Binomial[n+6,6], {n,0,30}] (* G. C. Greubel, Jul 21 2019 *)

vector(30, n, n--; 4^n*binomial(n+6,6) ) \\ G. C. Greubel, Jul 21 2019

[lucas_number2(n, 4, 0)*binomial(n,6)/2^12 for n in range(6, 36)] # Zerinvary Lajos, Mar 11 2009

a(n) = binomial(n+6, 6)*4^n.
G.f.: 1/(1 - 4*x)^7.
a(n) = A054335(n+13, 13).
E.g.f.: (45 + 1080*x + 5400*x^2 + 9600*x^3 + 7200*x^4 + 2304*x^5 + 256*x^6)*exp(4*x)/45. - G. C. Greubel, Jul 21 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 8394/5 - 5832*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 75000*log(5/4) - 83674/5. (End)

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A038845 3-fold convolution of A000302 (powers of 4).

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A038846 4-fold convolution of A000302 (powers of 4); expansion of g.f. 1/(1-4*x)^4.

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A040075 5-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^5.

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A163912 Least common multiple of all cycle sizes in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

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A163910 Number of cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

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A163911 Maximum cycle size in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

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A045543 6-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^6.

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A163914 Number of 3-cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

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