A081143 -id:A081143 - cp's OEIS frontend

A081144 Starting at 1, four-fold convolution of A000400 (powers of 6).

0, 0, 0, 1, 24, 360, 4320, 45360, 435456, 3919104, 33592320, 277136640, 2217093120, 17293326336, 132058128384, 990435962880, 7313988648960, 53287631585280, 383670947414016, 2733655500324864, 19296391766999040, 135074742368993280

Offset: 0

Paul Barry, Mar 08 2003

With a different offset, number of n-permutations (n=4) of 7 objects: t, u, v, w, z, x, y with repetition allowed, containing exactly three u's. Example: a(4)=24 because we have uuut, uutu, utuu, tuuu, uuuv, uuvu, uvuu, vuuu, uuuw, uuwu, uwuu, wuuu, uuuz, uuzu, uzuu, zuuu, uuux, uuxu, uxuu, xuuu, uuuy, uuyu, uyuu, yuuu. - Zerinvary Lajos, Jun 03 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-216,864,-1296).

Cf. A081143, A081136.

Cf. A038255.

List([-3..18],n->Binomial(n+3,3)*6^n); # Muniru A Asiru, Feb 19 2018

[6^n* Binomial(n+3, 3): n in [-3..20]]; // Vincenzo Librandi, Oct 16 2011

seq(seq(binomial(i+2, j)*6^(i-1), j =i-1), i=-2..19); # Zerinvary Lajos, Dec 30 2007
seq(binomial(n+3,3)*6^n,n=-3..18); # Zerinvary Lajos, Jun 03 2008

[lucas_number2(n, 6, 0)*binomial(n,3)/6^3 for n in range(0, 22)] # Zerinvary Lajos, Mar 13 2009

G.f.: x^3/(1 - 6*x)^4.
a(n) = 24*a(n-1) - 216*a(n-2) + 864*a(n-3) - 1296*a(n-4) for n > 3, a(0) = a(1) = a(2) = 0, a(3) = 1.
a(n) = 6^(n - 3)*binomial(n, 3).
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=3} 1/a(n) = 450*log(6/5) - 81.
Sum_{n>=3} (-1)^(n+1)/a(n) = 882*log(7/6) - 135. (End)

A128963 a(n) = (n^3 - n)*5^n.

Original entry on oeis.org

0, 150, 3000, 37500, 375000, 3281250, 26250000, 196875000, 1406250000, 9667968750, 64453125000, 418945312500, 2666015625000, 16662597656250, 102539062500000, 622558593750000, 3735351562500000, 22178649902343750, 130462646484375000, 761032104492187500

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A000351, A007531, A036289, A081143, A128796.

Cf. A128960, A128961, A128962, A128964, A128965, A128967, A128969.

[(n^3-n)*5^n: n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

Table[(n^3-n)5^n,{n,20}] (* or *) LinearRecurrence[{20,-150,500,-625},{0,150,3000,37500},20] (* Harvey P. Dale, Jul 22 2012 *)
CoefficientList[Series[150 x/(1 - 5 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)

a(1)=0, a(2)=150, a(3)=3000, a(4)=37500, a(n)=20*a(n-1)-150*a(n-2)+ 500*a(n-3)- 625*a(n-4). - Harvey P. Dale, Jul 22 2012
G.f.: 150*x^2/(1 - 5*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 150*A081143(n+1). - Bruno Berselli, Feb 12 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000351(n).
Sum_{n>=2} 1/a(n) = (8/5)*log(5/4) - 7/20.
Sum_{n>=2} (-1)^n/a(n) = (18/5)*log(6/5) - 13/20. (End)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

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

Original entry on oeis.org

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

A218016 Triangle, read by rows, where T(n,k) = k!C(n, k)5^(n-k) for n>=0, k=0..n.

Original entry on oeis.org

1, 5, 1, 25, 10, 2, 125, 75, 30, 6, 625, 500, 300, 120, 24, 3125, 3125, 2500, 1500, 600, 120, 15625, 18750, 18750, 15000, 9000, 3600, 720, 78125, 109375, 131250, 131250, 105000, 63000, 25200, 5040, 390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=5.
Sum(T(n,k), k=0..n) = A080954(n) (see the Formula section of A080954). . Also:
first column:    A000351;
second column:   A053464;
third column:  2*A084902;
fourth column: 6*A081143.

			Triangle begins:
1;
5,      1;
25,     10,     2;
125,    75,     30,     6;
625,    500,    300,    120,     24;
3125,   3125,   2500,   1500,    600,     120;
15625,  18750,  18750,  15000,   9000,    3600,   720;
78125,  109375, 131250, 131250,  105000,  63000,  25200,  5040;
390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000351, A053464, A080954, A081143, A084902, A090802, A217629, A218017.

[Factorial(n)/Factorial(n-k)*5^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*5^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 5^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(5x)*x^k.

A305837 Triangle read by rows: T(0,0) = 1; T(n,k) = 5*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 5, 25, 1, 125, 10, 625, 75, 1, 3125, 500, 15, 15625, 3125, 150, 1, 78125, 18750, 1250, 20, 390625, 109375, 9375, 250, 1, 1953125, 625000, 65625, 2500, 25, 9765625, 3515625, 437500, 21875, 375, 1, 48828125, 19531250, 2812500, 175000, 4375, 30, 244140625, 107421875, 17578125, 1312500, 43750, 525, 1

Offset: 0

Shara Lalo, Jun 11 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013612 ((1+5*x)^n).
The coefficients in the expansion of 1/(1-5x-x^2) are given by the sequence generated by the row sums.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 5.1925824035..., a metallic mean (see A098318), when n approaches infinity.

			Triangle begins:
            1;
            5;
           25,           1;
          125,          10;
          625,          75,          1;
         3125,         500,         15;
        15625,        3125,        150,         1;
        78125,       18750,       1250,        20;
       390625,      109375,       9375,       250,        1;
      1953125,      625000,      65625,      2500,       25;
      9765625,     3515625,     437500,     21875,      375,      1;
     48828125,    19531250,    2812500,    175000,     4375,     30;
    244140625,   107421875,   17578125,   1312500,    43750,    525,     1;
   1220703125,   585937500,  107421875,   9375000,   393750,   7000,    35;
   6103515625,  3173828125,  644531250,  64453125,  3281250,  78750,   700,  1;
  30517578125, 17089843750, 3808593750, 429687500, 25781250, 787500, 10500, 40;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 92, 380, 382.

Shara Lalo, Left-justified triangle
Shara Lalo, Skew diagonals in triangle A013612

Row sums give A052918.
Cf. A000351 (column 0), A053464 (column 1), A081135 (column 2), A081143 (column 3), A036071 (column 4).
Cf. A013612.
Cf. A098318.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 5 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten

G.f.: 1/(1 - 5*t*x - t^2).

A129005 a(n) = (n^3 + n^2)*5^n.

Original entry on oeis.org

0, 10, 300, 4500, 50000, 468750, 3937500, 30625000, 225000000, 1582031250, 10742187500, 70898437500, 457031250000, 2888183593750, 17944335937500, 109863281250000, 664062500000000, 3968811035156250, 23483276367187500

Offset: 0

Mohammad K. Azarian, May 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A036289, A081143, A128796.

[(n^3+n^2)*5^n: n in [0..25]]; // Vincenzo Librandi, Feb 12 2013

I:=[0,10,300,4500]; [n le 4 select I[n] else 20*Self(n-1)-150*Self(n-2)+500*Self(n-3)-625*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

CoefficientList[Series[10 x (1 + 10 x)/(1 - 5 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 12 2013 *)
Table[(n^3+n^2)5^n,{n,0,30}] (* or *) LinearRecurrence[{20,-150,500,-625},{0,10,300,4500},30] (* Harvey P. Dale, May 15 2022 *)

A129005(n)=5^n*(1+n)*n^2 \\ - M. F. Hasler, Feb 12 2013

G.f.: 10*x*(1 + 10*x)/(1 - 5*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 20*a(n-1)-150*a(n-2)+500*a(n-3)-625*a(n-4). - Vincenzo Librandi, Feb 12 2013
a(n) = 10*A081143(n+2)+100*A081143(n+1). - Bruno Berselli, Feb 13 2013

Initial term a(0)=0 added by M. F. Hasler, Feb 12 2013

A173113 a(n) = binomial(n + 10, 10) * 5^n.

Original entry on oeis.org

1, 55, 1650, 35750, 625625, 9384375, 125125000, 1519375000, 17092968750, 180425781250, 1804257812500, 17222460937500, 157872558593750, 1396564941406250, 11970556640625000, 99754638671875000

Offset: 0

Zerinvary Lajos, Feb 10 2010

With a different offset, number of n-permutations (n>=10) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly ten (10) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (55,-1375,20625,-206250,1443750,-7218750,25781250,-64453125,107421875,-107421875,48828125).

Cf. A053464, A084902, A081143, A036071, A050982, A140405, A140220, A140346, A140520.

[5^n*Binomial(n+10, 10): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Table[Binomial[n + 10, 10]*5^n, {n, 0, 20}]

a(n) = C(n + 10, 10)*5^n, n>=0.
G.f.: 1/(1-5*x)^11. - Vincenzo Librandi, Oct 15 2011
From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 184261655/63 - 13107200*log(5/4).
Sum_{n>=0} (-1)^n/a(n) = 503884800*log(6/5) - 11575501585/126. (End)

cp's OEIS Frontend

A081144 Starting at 1, four-fold convolution of A000400 (powers of 6).

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A128963 a(n) = (n^3 - n)*5^n.

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

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A218016 Triangle, read by rows, where T(n,k) = k!*C(n, k)*5^(n-k) for n>=0, k=0..n.

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A305837 Triangle read by rows: T(0,0) = 1; T(n,k) = 5*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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A129005 a(n) = (n^3 + n^2)*5^n.

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A173113 a(n) = binomial(n + 10, 10) * 5^n.

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A218016 Triangle, read by rows, where T(n,k) = k!C(n, k)5^(n-k) for n>=0, k=0..n.