A027473 -id:A027473 - cp's OEIS frontend

A053539 a(n) = n * 8^(n-1).

0, 1, 16, 192, 2048, 20480, 196608, 1835008, 16777216, 150994944, 1342177280, 11811160064, 103079215104, 893353197568, 7696581394432, 65970697666560, 562949953421312, 4785074604081152, 40532396646334464, 342273571680157696, 2882303761517117440, 24211351596743786496

Offset: 0

Barry E. Williams, Jan 15 2000

The Szeged index of the hypercube Q_n (see the Ashrafi et al. reference, p. 45, last line). - Emeric Deutsch, Aug 06 2014

For n > 3, 2*a(n) is the number of spanning trees in a superprism on 2*n vertices (see Bogdanowicz). - Stefano Spezia, May 05 2024

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. R. Ashrafi, B. Manoochehrian, and H. Yousefi-Azari, On Szeged polynomial of a graph, Bull. Iranian Math. Soc., 33, 2007, 37-46. - _Emeric Deutsch_, Aug 06 2014
Zbigniew R. Bogdanowicz, The number of spanning trees in a superprism, Discrete Math. Lett. 13 (2024) 66-73. See Theorem 3.1.
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Binomial transform of A027473.

Cf. A001787, A053464, A053469, A053540.

List([0..20], n-> n*8^(n-1)); # G. C. Greubel, May 16 2019

[n*8^(n-1): n in [0..20]]; // Vincenzo Librandi, Feb 09 2011

a := proc(n) option remember; if n<2 then n else 16*a(n-1)-64*a(n-2) end if end proc: seq(a(n), n = 0 .. 20); # Emeric Deutsch, Aug 06 2014

Table[n 8^(n-1),{n,0,20}] (* or *) LinearRecurrence[{16,-64},{0,1},20] (* Harvey P. Dale, Feb 01 2017 *)

a(n) = n*8^(n-1); \\ Joerg Arndt, Aug 07 2014

[n*8^(n-1) for n in (0..20)] # G. C. Greubel, May 16 2019

a(n) = 16*a(n-1) - 64*a(n-2), with a(0)=0, a(1)=1. - Emeric Deutsch, Aug 06 2014
From G. C. Greubel, May 16 2019: (Start)
G.f.: x/(1-8*x)^2.
E.g.f.: x*exp(8*x). (End)
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 8*log(8/7).
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(9/8). (End)

Offset corrected and name edited by Emeric Deutsch, Aug 06 2014

A212701 Main transitions in systems of n particles with spin 3.

Original entry on oeis.org

6, 84, 882, 8232, 72030, 605052, 4941258, 39530064, 311299254, 2421216420, 18643366434, 142367525496, 1079620401678, 8138676874188, 61040076556410, 455765904954528, 3389758918099302, 25124095510618356, 185639150161791186, 1367867422244777160, 10053825553499112126

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=7 (see formula), corresponding to spin S=(b-1)/2=3.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A001787, A212697, A212698, A212699, A212700, A212702, A212703, A212704 (b = 2, 3, 4, 5, 6, 8, 9, 10).

Cf. A000420, A008588, A027473.

LinearRecurrence[{14,-49},{6,84},20] (* Harvey P. Dale, Aug 02 2016 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212701.txt", n, " ", mtrans(n, 7)))

Vec(6*x/(7*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=7.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 14*a(n-1) - 49*a(n-2) for n > 2.
G.f.: 6*x/(7*x-1)^2. (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 6*x*exp(7*x).
a(n) = 6*A027473(n) = A008588(n)*A000420(n-1). (End)

A218017 Triangle, read by rows, where T(n,k) = k!C(n, k)7^(n-k) for n>=0, k=0..n.

Original entry on oeis.org

1, 7, 1, 49, 14, 2, 343, 147, 42, 6, 2401, 1372, 588, 168, 24, 16807, 12005, 6860, 2940, 840, 120, 117649, 100842, 72030, 41160, 17640, 5040, 720, 823543, 823543, 705894, 504210, 288120, 123480, 35280, 5040, 5764801, 6588344, 6588344, 5647152, 4033680, 2304960, 987840, 282240, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=7. Also:
first column:    A000420;
second column:   A027473;
third column:  2*A027474;
fourth column: 6*A140107.

			Triangle begins:
1;
7,       1;
49,      14,      2;
343,     147,     42,      6;
2401,    1372,    588,     168,     24;
16807,   12005,   6860,    2940,    840,     120;
117649,  100842,  72030,   41160,   17640,   5040,    720;
823543,  823543,  705894,  504210,  288120,  123480,  35280,  5040; etc.

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

Cf. A000420, A027466, A027473, A027474, A090802, A140107, A217629, A218016.

[Factorial(n)/Factorial(n-k)*7^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*7^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 7^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(7x)*x^k.

A104002 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern and containing it exactly once.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 12, 6, 1, 5, 32, 27, 8, 1, 6, 80, 108, 48, 10, 1, 7, 192, 405, 256, 75, 12, 1, 8, 448, 1458, 1280, 500, 108, 14, 1, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 10, 2304, 17496, 28672, 18750, 6480, 1372, 192, 18, 1, 11, 5120, 59049, 131072

Offset: 2

Ralf Stephan, Feb 26 2005

T(n+k,k+1) = total number of occurrences of any given letter in all possible n-length words on a k-letter alphabet. For example, with the 2 letter alphabet {0,1} there are 4 possible 2-length words: {00,01,10,11}. The letter 0 occurs 4 times altogether, as does the letter 1. T(4,3) = 4. - Ross La Haye, Jan 03 2007

Table T(n,k) = k*n^(k-1) n,k > 0 read by antidiagonals. - Boris Putievskiy, Dec 17 2012

			Triangle begins:
  1;
  2,   1;
  3,   4,    1;
  4,  12,    6,    1;
  5,  32,   27,    8,   1;
  6,  80,  108,   48,  10,   1;
  7, 192,  405,  256,  75,  12,  1;
  8, 448, 1458, 1280, 500, 108, 14, 1;

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2 <= n <= 150).
T. Mansour, Permutations containing and avoiding certain patterns, arXiv:math/9911243 [math.CO], 1999-2000.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

Cf. Left-edge columns include A001787, A027471, A002697, A053464, A053469, A027473, A053539, A053540, A053541, A081127, A081128.

Table[(n - k + 1) (k - 1)^(n - k), {n, 2, 12}, {k, 2, n}] // Flatten (* Michael De Vlieger, Aug 22 2018 *)

T(n, k) = (n-k+1) * (k-1)^(n-k), k<=n.

As a linear array, the sequence is a(n) = A004736(n)*A002260(n)^(A004736(n)-1) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2))^((t*t+3*t+4)/2-n-1), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

A230539 a(n) = 3n2^(3*n-1).

Original entry on oeis.org

0, 12, 192, 2304, 24576, 245760, 2359296, 22020096, 201326592, 1811939328, 16106127360, 141733920768, 1236950581248, 10720238370816, 92358976733184, 791648371998720, 6755399441055744, 57420895248973824, 486388759756013568, 4107282860161892352

Offset: 0

Bruno Berselli, Oct 23 2013

Arithmetic derivative of 8^n: a(n) = A003415(8^n).

Sum of reciprocals of a(n), for n>0: (2/3)*log(8/7).

Bruno Berselli, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018, A003415.
Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), this sequence, A230540 (k=9), A085708 (k=10), A081127 (k=11).
Row n=8 of A258997.

Magma
```
[3*n*2^(3*n-1): n in [0..20]];
```

A230539:=n->3*n*2^(3*n-1): seq(A230539(n), n=0..30); # Wesley Ivan Hurt, May 03 2017

Table[3 n 2^(3 n - 1), {n,0,20}]
LinearRecurrence[{16,-64},{0,12},20] (* Harvey P. Dale, Dec 25 2022 *)

a(n) = 3*n*2^(3*n-1); \\ Michel Marcus, Oct 23 2013

G.f.: 12*x/(1-8*x)^2.

a(n) = 12*A053539(n).

A230540 a(n) = 2n3^(2*n-1).

Original entry on oeis.org

0, 6, 108, 1458, 17496, 196830, 2125764, 22320522, 229582512, 2324522934, 23245229340, 230127770466, 2259436291848, 22029503845518, 213516729579636, 2058911320946490, 19765548681086304, 189008059262887782, 1801135623563989452, 17110788423857899794

Offset: 0

Bruno Berselli, Oct 23 2013

Arithmetic derivative of 9^n: a(n) = A003415(9^n).

Sum of reciprocals of a(n), for n>0: (3/2)*log(9/8).

Bruno Berselli, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A001019, A003415.

Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), A230539 (k=8), this sequence, A085708 (k=10), A081127 (k=11).

Magma
```
[2*n*3^(2*n-1): n in [0..20]];
```
Mathematica
```
Table[2 n 3^(2 n - 1), {n, 0, 20}]
```

a(n) = 2*n*3^(2*n-1); \\ Michel Marcus, Oct 23 2013

G.f.: 6*x/(1-9*x)^2.

a(n) = 6*A053540(n), with A053540(0)=0.

A317014 Triangle read by rows: T(0,0) = 1; T(n,k) = 7 * 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, 7, 49, 1, 343, 14, 2401, 147, 1, 16807, 1372, 21, 117649, 12005, 294, 1, 823543, 100842, 3430, 28, 5764801, 823543, 36015, 490, 1, 40353607, 6588344, 352947, 6860, 35, 282475249, 51883209, 3294172, 84035, 735, 1, 1977326743, 403536070, 29647548, 941192, 12005, 42

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 A013614 ((1+7*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A027466 ((7+x)^n).
The coefficients in the expansion of 1/(1-7x-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 7.14005494464025913554... ((7+sqrt(53))/2), a metallic mean (see A176439), when n approaches infinity.

			Triangle begins:
1;
7;
49, 1;
343, 14;
2401, 147, 1;
16807, 1372, 21;
117649, 12005, 294, 1;
823543, 100842, 3430, 28;
5764801, 823543, 36015, 490, 1;
40353607, 6588344, 352947, 6860, 35;
282475249, 51883209, 3294172, 84035, 735, 1;
1977326743, 403536070, 29647548, 941192, 12005, 42;
13841287201, 3107227739, 259416045, 9882516, 168070, 1029, 1;
96889010407, 23727920916, 2219448385, 98825160, 2117682, 19208, 49;
678223072849, 179936733613, 18643366434, 951192165, 24706290, 302526, 1372, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 96.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 7x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (7 + x)^n

Row sums give A054413.
Cf. A013614, A027466, A176439.
Cf. A000420 (column 0), A027473 (column 1), A027474 (column 2), A140107 (column 3), A139641 (column 4).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 7 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, 7*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

A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 12, 6, 1, 0, 0, 5, 32, 27, 8, 1, 0, 0, 6, 80, 108, 48, 10, 1, 0, 0, 7, 192, 405, 256, 75, 12, 1, 0, 0, 8, 448, 1458, 1280, 500, 108, 14, 1, 0, 0, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 0, 0, 10, 2304

Offset: 0

Franck Maminirina Ramaharo, Oct 14 2018

T(n,k) is the number of length n*k binary words of n consecutive blocks of length k, respectively, one of the blocks having exactly k letters 1, and the other having exactly one letter 0. First column follows from the next definition.
In Kauffman's language, T(n,k) is the total number of Jordan trails that are obtained by placing state markers at the crossings of the Pretzel universe P(k, k, ..., k) having n tangles, of k half-twists respectively. In other words, T(n,k) is the number of ways of splitting the crossings of the Pretzel knot shadow P(k, k, ..., k) such that the final diagram is a single Jordan curve. The aforementionned binary words encode these operations by assigning each tangle a length k binary words with the adequate choice for splitting the crossings.
Columns are linear recurrence sequences with signature (2*k, -k^2).

			Square array begins:
    0, 0,   0,    0,     0,      0,      0,      0, ...
    1, 1,   1,    1,     1,      1,      1,      1, ...
    0, 2,   4,    6,     8,     10,     12,     14, ... A005843
    0, 3,  12,   27,    48,     75,    108,    147, ... A033428
    0, 4,  32,  108,   256,    500,    864,   1372, ... A033430
    0, 5,  80,  405,  1280,   3125,   6480,  12005, ... A269792
    0, 6, 192, 1458,  6144,  18750,  46656, 100842, ...
    0, 7, 448, 5103, 28672, 109375, 326592, 823543, ...
    ...
T(3,2) = 3*2^(3 - 1) = 12. The corresponding binary words are 110101, 110110, 111001, 111010, 011101, 011110, 101101, 101110, 010111, 011011, 100111, 101011.

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Alexander Stoimenow, Everywhere Equivalent 2-Component Links, Symmetry Vol. 7 (2015), 365-375.
Wikipedia, Pretzel link

Antidiagonal sums: A101495.
Column 1 is column 2 of A300453.
Column 2 is column 1 of A300184.
Cf. A104002, A320530.

T[n_, k_] = If [k > 0, n*k^(n - 1), If[k == 0 && n == 1, 1, 0]];
Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 12}]//Flatten

T(n, k) := if k > 0 then n*k^(n - 1) else if k = 0 and n = 1 then 1 else 0$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, nn))$

T(n,k) = (2*k)*T(n-1,k) - (k^2)*T(n-2,k).
G.f. for columns: x/(1 - k*x)^2.
E.g.f. for columns: x*exp(k*x).
T(n,1) = A001477(n).
T(n,2) = A001787(n).
T(n,3) = A027471(n+1).
T(n,4) = A002697(n).
T(n,5) = A053464(n).
T(n,6) = A053469(n), n > 0.
T(n,7) = A027473(n), n > 0.
T(n,8) = A053539(n).
T(n,9) = A053540(n), n > 0.
T(n,10) = A053541(n), n > 0.
T(n,11) = A081127(n).
T(n,12) = A081128(n).

cp's OEIS Frontend

A053539 a(n) = n * 8^(n-1).

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A212701 Main transitions in systems of n particles with spin 3.

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A218017 Triangle, read by rows, where T(n,k) = k!*C(n, k)*7^(n-k) for n>=0, k=0..n.

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A104002 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern and containing it exactly once.

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A230539 a(n) = 3*n*2^(3*n-1).

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A230540 a(n) = 2*n*3^(2*n-1).

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A317014 Triangle read by rows: T(0,0) = 1; T(n,k) = 7 * 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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A218017 Triangle, read by rows, where T(n,k) = k!C(n, k)7^(n-k) for n>=0, k=0..n.

A230539 a(n) = 3n2^(3*n-1).

A230540 a(n) = 2n3^(2*n-1).