A038231 -id:A038231 - cp's OEIS frontend

A038243 Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).

1, 5, 1, 25, 10, 1, 125, 75, 15, 1, 625, 500, 150, 20, 1, 3125, 3125, 1250, 250, 25, 1, 15625, 18750, 9375, 2500, 375, 30, 1, 78125, 109375, 65625, 21875, 4375, 525, 35, 1, 390625, 625000, 437500, 175000, 43750, 7000, 700, 40, 1, 1953125, 3515625, 2812500, 1312500, 393750, 78750, 10500, 900, 45, 1

Offset: 0

N. J. A. Sloane

Mirror image of A013612. - Zerinvary Lajos, Nov 25 2007
T(i,j) is the number of i-permutations of 6 objects a,b,c,d,e,f, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007
Triangle of coefficients in expansion of (5+x)^n - N-E. Fahssi, Apr 13 2008
Also the convolution triangle of A000351. - Peter Luschny, Oct 09 2022

			Triangle begins as:
       1;
       5,      1;
      25,     10,      1;
     125,     75,     15,      1;
     625,    500,    150,     20,     1;
    3125,   3125,   1250,    250,    25,    1;
   15625,  18750,   9375,   2500,   375,   30,   1;
   78125, 109375,  65625,  21875,  4375,  525,  35,  1;
  390625, 625000, 437500, 175000, 43750, 7000, 700, 40, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
B. N. Cyvin, J. Brunvoll, and S. J. Cyvin, Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), 109-121.

Cf. A000351, A000400 (row sums), A036071, A050982, A053464, A081135, A081143.

Sequences of the form q^(n-k)*binomial(n, k): A007318 (q=1), A038207 (q=2), A027465 (q=3), A038231 (q=4), this sequence (q=5), A038255 (q=6), A027466 (q=7), A038279 (q=8), A038291 (q=9), A038303 (q=10), A038315 (q=11), A038327 (q=12), A133371 (q=13), A147716 (q=14), A027467 (q=15).

[5^(n-k)*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 12 2021

for i from 0 to 8 do seq(binomial(i, j)*5^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 5^(n-1)); # Peter Luschny, Oct 09 2022

With[{q=5}, Table[q^(n-k)*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, May 12 2021 *)

flatten([[5^(n-k)*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 12 2021

See A038207 and A027465 and replace 2 and 3 in analogous formulas with 5. - Tom Copeland, Oct 26 2012

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)

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)

A048786 Triangle of coefficients of certain exponential convolution polynomials.

Original entry on oeis.org

1, 8, 1, 96, 24, 1, 1536, 576, 48, 1, 30720, 15360, 1920, 80, 1, 737280, 460800, 76800, 4800, 120, 1, 20643840, 15482880, 3225600, 268800, 10080, 168, 1, 660602880, 578027520, 144506880, 15052800, 752640, 18816, 224, 1

Offset: 1

Wolfdieter Lang

i) p(n,x) := sum(a(n,m)*x^m,m=1..n), p(0,x) := 1, are monic polynomials satisfying p(n,x+y)= sum(binomial(n,k)*p(k,x)*p(n-k,y),k=0..n), (exponential convolution polynomials). ii) In the terminology of the umbral calculus (see reference) p(n,x) are called associated to f(t)= t/(1+4*t). iii) a(n,1)= A034177(n).
Also the Bell transform of A034177. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016
Also the fourth power of the unsigned Lah triangular matrix A105278. - Shuhei Tsujie, May 18 2019
Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -3 <= d <= 4). - Shuhei Tsujie, May 18 2019

			Triangle begins:
      1;
      8,     1;
     96,    24,    1;
   1536,   576,   48,  1;
  30720, 15360, 1920, 80, 1;
  ...

S. Roman, The Umbral Calculus, Academic Press, New York, 1984

N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A034177, A038231, A008297.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 4^n*(n+1)!, 9); # Peter Luschny, Jan 28 2016

rows = 8;
t = Table[4^n*(n+1)!, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

a(n, m) = n!*4^(n-m)*binomial(n-1, m-1)/m!, n >= m >= 1; a(n, m) := 0, m>n; a(n, m) = (n!/m!)*A038231(n-1, m-1) = 4^(n-m)*A008297(n, m) (Lah-triangle).

T(8,4) corrected by Jean-François Alcover, Jun 22 2018

A111959 Renewal array for aerated central binomial coefficients.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 6, 0, 6, 0, 1, 0, 16, 0, 8, 0, 1, 20, 0, 30, 0, 10, 0, 1, 0, 64, 0, 48, 0, 12, 0, 1, 70, 0, 140, 0, 70, 0, 14, 0, 1, 0, 256, 0, 256, 0, 96, 0, 16, 0, 1, 252, 0, 630, 0, 420, 0, 126, 0, 18, 0, 1, 0, 1024, 0, 1280, 0, 640, 0, 160, 0, 20, 0, 1, 924, 0, 2772, 0

Offset: 0

Paul Barry, Aug 23 2005

Row sums are A098615.
Binomial transform (product with C(n,k)) is A111960.
Diagonal sums are A026671 (with interpolated zeros).
Inverse is (1/sqrt(1+4x^2),x/sqrt(1+4x^2)), or (sqrt(-1))^(n-k)*T(n,k). [corrected by Peter Bala, Aug 13 2021]
The Riordan array (1,x/sqrt(1-4*x^2)) is the same array with an additional column of zeros (besides the top element 1) added to the left. - Vladimir Kruchinin, Feb 17 2011

			From _Peter Bala_, Aug 13 2021: (Start)
Triangle begins
  1;
  0,  1;
  2,  0, 1;
  0,  4, 0, 1;
  6,  0, 6, 0, 1;
  0, 16, 0, 8, 0, 1;
Infinitesimal generator begins
  0;
  0, 0;
  2, 0, 0;
  0, 4, 0, 0;
  0, 0, 6, 0, 0;
  0, 0, 0, 8, 0, 0; (End)

Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A054335, A038231, A046521.

Riordan array (1/sqrt(1-4x^2), x/sqrt(1-4x^2)); number triangle T(n, k)=(1+(-1)^(n-k))*binomial((n-1)/2, (n-k)/2)*2^(n-k)/2.
G.f.: 1/(1-xy-2x^2/(1-x^2/(1-x^2/(1-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
From Peter Bala, Aug 13 2021: (Start)
T(2*n,2*k) = A046521(n,k); T(2*n+1,2*k+1) = A038231(n,k).
The row entries, read from right to left, are the coefficients in the n-th order Taylor polynomial of (sqrt(1 + 4*x^2))^((n-1)/2) at x = 0.
The infinitesimal generator of this array has the sequence [2, 4, 6, 8, 10, ...] on the second subdiagonal below the main diagonal and zeros elsewhere.
The m-th power of the array is the Riordan array (1/sqrt(1 - 4*m*x^2), x/sqrt(1 - 4*m*x^2)) with entries given by sqrt(m)^(n-k)*T(n,k). (End)

A147716 Triangle of coefficients in expansion of (14 + x)^n.

Original entry on oeis.org

1, 14, 1, 196, 28, 1, 2744, 588, 42, 1, 38416, 10976, 1176, 56, 1, 537824, 192080, 27440, 1960, 70, 1, 7529536, 3226944, 576240, 54880, 2940, 84, 1, 105413504, 52706752, 11294304, 1344560, 96040, 4116, 98, 1, 1475789056, 843308032, 210827008, 30118144, 2689120, 153664, 5488, 112, 1

Offset: 0

Philippe Deléham, Nov 11 2008

Triangle T(n,k), read by rows, given by [14, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Triangle begins :
       1;
      14,      1;
     196,     28,     1;
    2744,    588,    42,    1;
   38416,  10976,  1176,   56,  1;
  537824, 192080, 27440, 1960, 70, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001024, A023531, A130595.

Sequences of the form q^(n-k)*binomial(n, k): A007318 (q=1), A038207 (q=2), A027465 (q=3), A038231 (q=4), A038243 (q=5), A038255 (q=6), A027466 (q=7), A038279 (q=8), A038291 (q=9), A038303 (q=10), A038315 (q=11), A038327 (q=12), A133371 (q=13), this sequence (q=14), A027467 (q=15).

[14^(n-k)*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 15 2021

With[{m=8}, CoefficientList[CoefficientList[Series[1/(1-14*x-x*y), {x, 0, m}, {y, 0, m}], x], y]]//Flatten (* Georg Fischer, Feb 17 2020 *)

flatten([[14^(n-k)*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 15 2021

T(n,k) = binomial(n,k) * 14^(n-k).
G.f.: 1/(1 - 14*x - x*y). - R. J. Mathar, Aug 12 2015
Sum_{k=0..n} T(n, k) = 15^n = A001024(n). - G. C. Greubel, May 15 2021

a(36) corrected by Georg Fischer, Feb 17 2020

A054338 8-fold convolution of A000302 (powers of 4).

Original entry on oeis.org

1, 32, 576, 7680, 84480, 811008, 7028736, 56229888, 421724160, 2998927360, 20392706048, 133479530496, 845370359808, 5202279137280, 31213674823680, 183120225632256, 1052941297385472, 5946021444059136, 33033452466995200, 180814687187763200, 976399310813921280

Offset: 0

Wolfdieter Lang, Mar 13 2000

With a different offset, number of n-permutations (n>=7) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly seven (7) u's. - Zerinvary Lajos, Jun 23 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (32,-448,3584,-17920,57344,-114688,131072,-65536).

Cf. A000302, A054335.

Cf. A038231.

List([0..20], n-> 4^n*Binomial(n+7,7) ); # G. C. Greubel, Jul 21 2019

[4^n*Binomial(n+7, 7): n in [0..20]]; // Vincenzo Librandi, Oct 15 2011

seq(binomial(n+7,7)*4^n,n=0..20); # Zerinvary Lajos, Jun 23 2008

Table[4^n*Binomial[n+7,7], {n,0,20}] (* G. C. Greubel, Jul 21 2019 *)
LinearRecurrence[{32,-448,3584,-17920,57344,-114688,131072,-65536},{1,32,576,7680,84480,811008,7028736,56229888},30] (* Harvey P. Dale, Jun 08 2025 *)

vector(20, n, n--; 4^n*binomial(n+7,7)) \\ G. C. Greubel, Jul 21 2019

a(n) = binomial(n+7, 7)*4^n.
G.f.: 1/(1-4*x)^8.
a(n) = A054335(n+15, 15).
E.g.f.: (315 + 8820*x + 52920*x^2 + 117600*x^3 + 117600*x^4 + 56448*x^5 + 12544*x^6 + 1024*x^7)*exp(4*x)/315. - G. C. Greubel, Jul 21 2019
From Amiram Eldar, Mar 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 20412*log(4/3) - 88067/15.
Sum_{n>=0} (-1)^n/a(n) = 437500*log(5/4) - 292873/3. (End)

A054339 9-fold convolution of A000302 (powers of 4).

Original entry on oeis.org

1, 36, 720, 10560, 126720, 1317888, 12300288, 105431040, 843448320, 6372720640, 45883588608, 317013884928, 2113425899520, 13655982735360, 85837605765120, 526470648692736, 3158823892156416, 18581317012684800, 107358720517734400, 610249569258700800

Offset: 0

Wolfdieter Lang, Mar 13 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (36,-576,5376,-32256,129024,-344064,589824,-589824,262144).

Cf. A000302, A054335.

Cf. A038231.

List([0..20], n-> 4^n*Binomial(n+8, 8)); # G. C. Greubel, Jul 21 2019

[Binomial(n+8, 8)*4^n: n in [0..20]]; // Vincenzo Librandi, May 31 2011

seq(binomial(n+8,8)*4^n,n=0..20); # Zerinvary Lajos, Jun 23 2008

Table[Binomial[n+8,8]4^n,{n,0,20}] (* or *) LinearRecurrence[ {36,-576,5376,-32256,129024,-344064,589824,-589824,262144},{1,36,720,10560,126720,1317888,12300288,105431040,843448320},20]

vector(20, n, n--; 4^n*binomial(n+8, 8)) \\ G. C. Greubel, Jul 21 2019

[4^n*binomial(n+8, 8) for n in (0..20)] # G. C. Greubel, Jul 21 2019

a(n) = binomial(n+8, 8)*4^n.
G.f.: 1/(1-4*x)^9.
a(n) = A054335(n+17, 17).
a(n) = 36*a(n-1) - 576*a(n-2) + 5376*a(n-3) - 32256*a(n-4) + 129024*a(n-5) - 344064*a(n-6) + 589824*a(n-7) - 589824*a(n-8) + 262144*a(n-9). - Harvey P. Dale, Aug 30 2013
E.g.f.: (16/7!)*(315 + 10080*x + 70560*x^2 + 188160*x^3 + 235200*x^4 + 150528*x^5 + 50176*x^6 + 8192*x^7 + 512*x^8)*exp(4*x). - G. C. Greubel, Jul 21 2019
From Amiram Eldar, Mar 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 704696/35 - 69984*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 2500000*log(5/4) - 11715016/21. (End)

A141478 a(n) = binomial(n+2,3)*4^3.

Original entry on oeis.org

64, 256, 640, 1280, 2240, 3584, 5376, 7680, 10560, 14080, 18304, 23296, 29120, 35840, 43520, 52224, 62016, 72960, 85120, 98560, 113344, 129536, 147200, 166400, 187200, 209664, 233856, 259840, 287680, 317440, 349184, 382976, 418880, 456960, 497280, 539904, 584896

Offset: 1

Zerinvary Lajos, Aug 09 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A035008, A038231 (3rd subdiagonal), A210440.

[Binomial(n+2,3)*4^3: n in [1..34]];  // Bruno Berselli, Apr 07 2011

I:=[64, 256, 640, 1280]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 29 2012

Maple
```
seq(binomial(n+2,3)*4^3, n=1..36);
```

CoefficientList[Series[64/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 29 2012 *)

G.f.: 64*x/(1-x)^4.
a(n) = 32*n*(n+1)*(n+2)/3 = 64*A000292(n).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 29 2012
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=1} 1/a(n) = 3/128.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)/16 - 15/128. (End)
From Elmo R. Oliveira, Aug 19 2025: (Start)
E.g.f.: 32*x*(6 + 6*x + x^2)*exp(x)/3.
a(n) = 16*A210440(n). (End)

Offset adapted to the g.f. by Bruno Berselli, Apr 07 2011

A120054 a(n) = binomial(n+3,4)*4^4.

Original entry on oeis.org

256, 1280, 3840, 8960, 17920, 32256, 53760, 84480, 126720, 183040, 256256, 349440, 465920, 609280, 783360, 992256, 1240320, 1532160, 1872640, 2266880, 2720256, 3238400, 3827200, 4492800, 5241600, 6080256, 7015680, 8055040, 9205760, 10475520, 11872256, 13404160

Offset: 1

Zerinvary Lajos, Aug 07 2008

Number of n permutations (n>=4) of 5 objects u, v, z, x, y with repetition allowed, containing n-4 u's. Example: if n=4 then n-4 = zero (0) u, a(1)=256, if n=5 then n-4 = one (1) u, a(2)=1280, if n=6 then n-4 = two (2) u, a(3)=3840, etc.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A035008, A008586, A038231.

Maple
```
seq(binomial(n+3,4)*4^4, n=1..36);
```

256*Binomial[Range[30]+3,4] (* or *) LinearRecurrence[{5,-10,10,-5,1},{256,1280,3840,8960,17920},30] (* Harvey P. Dale, Jul 19 2018 *)

G.f.: 256/(1-x)^5.
a(n) = C(n+3,4)*4^4, n>=1.
From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/192.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/8 - 1/12. (End)

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A038243 Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).

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

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

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A048786 Triangle of coefficients of certain exponential convolution polynomials.

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A111959 Renewal array for aerated central binomial coefficients.

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A147716 Triangle of coefficients in expansion of (14 + x)^n.

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