A140802 -id:A140802 - cp's OEIS frontend

A053107 Expansion of 1/(1-8*x)^8.

1, 64, 2304, 61440, 1351680, 25952256, 449839104, 7197425664, 107961384960, 1535450808320, 20882130993152, 273366078455808, 3462636993773568, 42617070692597760, 511404848311173120, 6000483553517764608, 69005560865454292992, 779356922715719073792

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n>=7) of 9 objects: p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly 7 u's. - Zerinvary Lajos, Feb 11 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (64, -1792, 28672, -286720, 1835008, -7340032, 16777216, -16777216).

Cf. A036226.

Cf. A081138, A140802, A175210, A140406, A053107, A141054, A173155. - Zerinvary Lajos, Feb 11 2010

[8^n* Binomial(n+7, 7): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Table[Binomial[n + 7, 7]*8^n, {n, 0, 20}] (* Zerinvary Lajos, Feb 11 2010 *)
CoefficientList[Series[1/(1-8x)^8,{x,0,20}],x] (* or *) LinearRecurrence[ {64,-1792,28672,-286720,1835008,-7340032,16777216,-16777216},{1,64,2304,61440,1351680,25952256,449839104,7197425664},20] (* Harvey P. Dale, Jul 19 2018 *)

vector(30, n, n--; 8^n*binomial(n+7,7)) \\ G. C. Greubel, Aug 16 2018

[lucas_number2(n, 8, 0)*binomial(n,7)/8^7 for n in range(7, 22)] # Zerinvary Lajos, Mar 13 2009

a(n) = 8^n*binomial(n+7, 7).

G.f.: 1/(1-8*x)^8.

More terms from Harvey P. Dale, Jul 19 2018

A128967 a(n) = (n^3-n)*8^n.

Original entry on oeis.org

0, 384, 12288, 245760, 3932160, 55050240, 704643072, 8455716864, 96636764160, 1063004405760, 11338713661440, 117922622078976, 1200666697531392, 12006666975313920, 118219490218475520, 1148417904979476480, 11024811887802974208, 104735712934128254976

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 (32,-384,2048,-4096).

Cf. A001018, A007531, A036289, A128796, A140802.

Cf. A128960, A128961, A128962, A128963, A128964, A128965, A128969.

[(n^3 - n)*8^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

LinearRecurrence[{32, -384, 2048, -4096}, {0, 384, 12288, 245760}, 30] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 384x^2/(1-8x)^4.
a(n) = 384*A140802(n-2). (End)
a(n) = 32*a(n-1) - 384*a(n-2) + 2048*a(n-3) - 4096*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A001018(n).
Sum_{n>=2} 1/a(n) = (49/16)*log(8/7) - 13/32.
Sum_{n>=2} (-1)^n/a(n) = (81/16)*log(9/8) - 19/32. (End)

Corrected the offset. - Mohammad K. Azarian, Nov 20 2008

A111636 Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 96, 32, 1, 1, 80, 640, 640, 80, 1, 1, 192, 3840, 10240, 3840, 192, 1, 1, 448, 21504, 143360, 143360, 21504, 448, 1, 1, 1024, 114688, 1835008, 4587520, 1835008, 114688, 1024, 1, 1, 2304, 589824, 22020096, 132120576, 132120576, 22020096, 589824, 2304, 1

Offset: 0

Emeric Deutsch, Aug 09 2005

Row sums yield A047863. T(2*n,n) = A111637(n). T(n,1) = A001787(n).

			T(2,1)=4 because we have B G, B--G, G B and G--B, where B (G) stands for a blue (green) node and -- denotes an edge.
Triangle starts:
  1;
  1,  1;
  1,  4,  1;
  1, 12, 12,  1;
  1, 32, 96, 32, 1;
  ...

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 88, Eq. 3.11.2.

S. R. Finch, Bipartite, k-colorable and k-colored graphs
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A047863, A111637, A001787.
Cf. A134530 (matrix log), A134531.
Cf. A000684, A011266, A038845, A140802, A224069 (matrix inverse).

T:=(n,k)->binomial(n,k)*2^(k*(n-k)): for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

nn=6;f[x_,y_]:=Sum[Exp[x 2^n](y x)^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[f[x,y],{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Sep 07 2013 *)

T(n, k)=2^[k(n-k)]*C(n, k).
Matrix log yields triangle A134530, where A134530(n,k) = A134531(n-k)*(2^k)^(n-k)*C(n,k). - Paul D. Hanna, Nov 11 2007
From Peter Bala, Aug 13 2012: (Start)
Let f(n) = n!*2^binomial(n,2) = A011266(n). Then T(n,k) = f(n)/(f(k)*f(n-k)).
E.g.f.: Sum_{n >= 0} exp(2^n*t*x)*x^n/n! = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + ....
O.g.f.: Sum_{n >= 0} x^n/(1-2^n*t*x)^(n+1) = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... O.g.f. for column k: 1/(1-2^k*x)^(k+1).
Recurrence equation: T(n,k) = 2^k*T(n-1,k) + 2^(n-k)*T(n-1,k-1).
Column k = 2: A038845. Column k = 3: A140802. Sum_{k = 0..n} k*T(n,k) = n*A000684(n). (End)
From Peter Bala, Apr 09 2013: (Start)
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + .... Then a generating function for this sequence is E(z)*E(x*z) = 1 + (1 + x)*z + (1 + 4*x + x^2)*z^2/(2!*2) + (1 + 12*x + 12*x^2 + x^3)*z^3/(3!*2^3) + .... Cf. Pascal's triangle A007318 with an e.g.f. of exp(z)*exp(x*z).
This is a generalized Riordan array (E(x), x) with respect to the sequence n!*2^C(n,2), as defined by Wang and Wang.
The n-th power of this triangle has a generating function E(z)^n*E(x*z). See A224069 for the inverse array (n = -1).
The n-th row is a log-concave sequence and hence unimodal.
The row polynomials satisfy the recurrence equation R(n+1,x) = 2^n*x*R(n,x/2) + R(n,2*x) with R(0,x) = 1, as well as R'(n,2*x) = n*2^(n-1)*R(n-1,x) (the ' denotes differentiation w.r.t. x). The row polynomials appear to have only real zeros.
Sum_{k = 0..n} (-1)^k*T(2*n+1,k) = 0;
Sum_{k = 0..n} (-1)^k*2^k*T(2*n,k) = 0;
Sum_{k = 0..n} 2^k*T(n,k) = A000684(n). (End)
T(n,k+1) = Product_{i=0..k} (T(n-i,1)/T(i+1,1)) for 0 <= k < n. - Werner Schulte, Nov 13 2018

A173155 a(n) = binomial(n + 5, 5) * 8^n.

Original entry on oeis.org

1, 48, 1344, 28672, 516096, 8257536, 121110528, 1660944384, 21592276992, 268703891456, 3224446697472, 37520834297856, 425236122042368, 4710307813392384, 51140484831117312, 545498504865251328, 5727734301085138944, 59298896293587320832, 606166495445559279616

Offset: 0

Zerinvary Lajos, Feb 11 2010

With a different offset, number of n-permutations (n>=5) of 9 objects: p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly five (5) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (48,-960,10240,-61440,196608,-262144).

Cf. A081138, A140802, A175210, A140406, A053107, A141054.

[8^n* Binomial(n+5, 5): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Table[Binomial[n + 5, 5]*8^n, {n, 0, 20}]

a(n) = C(n + 5, 5)*8^n, n>=0.
G.f.: 1/(1-8*x)^6. - Vincenzo Librandi, Oct 16 2011
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 96040*log(8/7) - 38470/3.
Sum_{n>=0} (-1)^n/a(n) = 262440*log(9/8) - 30910. (End)

A172510 a(n) = binomial(n + 4, 4) * 8^n.

Original entry on oeis.org

1, 40, 960, 17920, 286720, 4128768, 55050240, 692060160, 8304721920, 95965675520, 1074815565824, 11725260718080, 125069447659520, 1308418837053440, 13458022323978240, 136374626216312832, 1363746262163128320, 13477021884906209280, 131775325096860712960

Offset: 0

Zerinvary Lajos, Feb 05 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..138
Index entries for linear recurrences with constant coefficients, signature (40,-640,5120,-20480,32768).

Cf. A081138, A140802, A140406, A053107, A141054.

[Binomial(n + 4, 4)*8^n: n in [0..30]]; // Vincenzo Librandi, Jun 06 2011

Table[Binomial[n + 4, 4]*8^n, {n, 0, 25}]

Vec(1 / (1-8*x)^5 + O(x^30)) \\ Colin Barker, Jul 24 2017

G.f.: 1 / (1-8*x)^5. - R. J. Mathar, Feb 11 2010
a(n) = (8^(-1 + n)*(1 + n)*(2 + n)*(3 + n)*(4 + n)) / 3. - Colin Barker, Jul 24 2017
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 4400/3 - 10976*log(8/7).
Sum_{n>=0} (-1)^n/a(n) = 23328*log(9/8) - 8240/3. (End)

A196280 a(n) = binomial(n+9, 9)*8^n.

Original entry on oeis.org

1, 80, 3520, 112640, 2928640, 65601536, 1312030720, 23991418880, 407854120960, 6525665935360, 99190122217472, 1442765414072320, 20198715797012480, 273459536944168960, 3594039628409077760, 46003707243636195328, 575046340545452441600, 7035861107850241638400

Offset: 0

Vincenzo Librandi, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (80,-2880,61440,-860160,8257536,-55050240,251658240,-754974720,1342177280,-1073741824).

Cf. A140406, A140802, A141054, A173155

Magma
```
[Binomial(n+9, 9)*8^n: n in [0..20]];
```

Table[Binomial[n+9,9]8^n,{n,0,20}] (* or *) LinearRecurrence[{80,-2880,61440,-860160,8257536,-55050240,251658240,-754974720,1342177280,-1073741824},{1,80,3520,112640,2928640,65601536,1312030720,23991418880,407854120960,6525665935360},20] (* Harvey P. Dale, May 13 2017 *)

a(n) = C(n+9,9)*8^n.
G.f.: 1 / (8*x-1)^10 . - R. J. Mathar, Oct 13 2011
From Amiram Eldar, Feb 17 2023: (Start)
Sum_{n>=0} 1/a(n) = 415065672*log(8/7) - 277121481/5.
Sum_{n>=0} (-1)^n/a(n) = 3099363912*log(9/8) - 12776837121/35. (End)

A317028 Triangle read by rows: T(0,0) = 1; T(n,k) = 8 * 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, 8, 64, 1, 512, 16, 4096, 192, 1, 32768, 2048, 24, 262144, 20480, 384, 1, 2097152, 196608, 5120, 32, 16777216, 1835008, 61440, 640, 1, 134217728, 16777216, 688128, 10240, 40, 1073741824, 150994944, 7340032, 143360, 960, 1, 8589934592, 1342177280, 75497472, 1835008, 17920, 48

Offset: 0

Zagros Lalo, Jul 19 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013615 ((1+8*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A038279 ((8+x)^n).
The coefficients in the expansion of 1/(1-8x-x^2) are given by the sequence generated by the row sums.
The row sums are Denominators of continued fraction convergents to sqrt(17), see A041025.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 8.12310562561766054982... (a metallic mean), when n approaches infinity (see A176458: (4+sqrt(17))).

			Triangle begins:
1;
8;
64, 1;
512, 16;
4096, 192, 1;
32768, 2048, 24;
262144, 20480, 384, 1;
2097152, 196608, 5120, 32;
16777216, 1835008, 61440, 640, 1;
134217728, 16777216, 688128, 10240, 40;
1073741824, 150994944, 7340032, 143360, 960, 1;
8589934592, 1342177280, 75497472, 1835008, 17920, 48;
68719476736, 11811160064, 754974720, 22020096, 286720, 1344, 1;
549755813888, 103079215104, 7381975040, 251658240, 4128768, 28672, 56;
4398046511104, 893353197568, 70866960384, 2768240640, 55050240, 516096, 1792, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, Pages 70, 98

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 8x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (8 + x)^n

Row sums give A041025.
Cf. A013615, A038279, A176458.
Cf. A001018 (column 0), A053539 (column 1), A081138 (column 2), A140802 (column 3), A172510 (column 4).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 8 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 8*T(n-1, k)+T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

A197321 a(n) = binomial(n+10, 10)*8^n.

Original entry on oeis.org

1, 88, 4224, 146432, 4100096, 98402304, 2099249152, 40785412096, 734137417728, 12398765277184, 198380244434944, 3029807369551872, 44437174753427456, 628956934971588608, 8625695108181786624, 115009268109090488320, 1495120485418176348160, 18996824991195652423680

Offset: 0

Vincenzo Librandi, Oct 15 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (88,-3520,84480,-1351680,15138816,-121110528,692060160,-2768240640,7381975040,-11811160064,8589934592).

Cf. A140406, A140802, A141054, A173155, A196280.

Magma
```
[8^n*Binomial(n+10, 10): n in [0..20]]
```

Table[Binomial[n+10,10]8^n,{n,0,20}] (* Harvey P. Dale, Mar 05 2012 *)

a(n) = 8^n*C(n+10, 10).
G.f.: 1/(1-8*x)^11.
From Amiram Eldar, Feb 17 2023: (Start)
Sum_{n>=0} 1/a(n) = 3879700814/9 - 3228288560*log(8/7).
Sum_{n>=0} (-1)^n/a(n) = 30993639120*log(9/8) - 229983068738/63. (End)

cp's OEIS Frontend

A053107 Expansion of 1/(1-8*x)^8.

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A128967 a(n) = (n^3-n)*8^n.

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A111636 Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.

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A173155 a(n) = binomial(n + 5, 5) * 8^n.

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A172510 a(n) = binomial(n + 4, 4) * 8^n.

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A196280 a(n) = binomial(n+9, 9)*8^n.

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A317028 Triangle read by rows: T(0,0) = 1; T(n,k) = 8 * 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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