A005061 -id:A005061 - cp's OEIS frontend

A069383 Number of n X 8 binary arrays with a path of adjacent 1's from top row to bottom row.

255, 58975, 12519345, 2541013617, 506544513343, 100193001007135, 19752116290408897, 3888492110032895921, 765068584835726823447, 150495460332124702700743, 29601436900513534956486417, 5822276945638780019957186657, 1145175890285973091225709069559

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069384 Number of n X 9 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

511, 242461, 105683341, 43843180113, 17792504911231, 7146486319870849, 2856738227918860537, 1139431807812280823353, 454016084146597129480639, 180827014727915133583355805, 72007238386097441302774324845, 28672084199736350940864453233905

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069385 Number of n X 10 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1023, 989527, 882516857, 746691527217, 615793150236223, 501394148790682463, 405671453894948801169, 327195886636045773364289, 263496356226918378172858791, 212041997127527144580159558415, 170577316159214009818521435822905, 137200626549758017615769211506956897

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069386 Number of n X 11 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

2047, 4017157, 7308428597, 12588144461329, 21067276157958271, 34728307983514473409, 56797093221062281961353, 92506527736225809457497129, 150343681421865609328377897039, 244072929303780515911965529991989, 396017759826921556057184566557279157

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..100

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069387 Number of n X 12 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

4095, 16245775, 60131384705, 210502738714097, 714097521397778495, 2380826437768134741023, 7862987107940658832635809, 25834416752721407468468718385, 84641600245355780352751022513463, 276889755885138056467643815501003447, 905061607197623234620822097612564300097

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..100

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A137786 a(n) = 4^n - 3^n - 2^n.

Original entry on oeis.org

-1, -1, 3, 29, 159, 749, 3303, 14069, 58719, 241949, 988503, 4015109, 16241679, 65506349, 263636103, 1059360149, 4251855039, 17050597949, 68331794103, 273715121189, 1096023794799, 4387584060749, 17560800790503, 70274592610229, 281192530396959, 1125052584678749

Offset: 0

Vladimir Joseph Stephan Orlovsky, Apr 28 2008

a(n) mod 100 = 49 for n = 4*k + 1, k > 0; a(n) mod 100 = 3 for n = 4*k + 2, k >= 0. [Alex Ratushnyak, Jul 03 2012]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Cf. A001047, A005061, A074526, A083324, A147975.

I:=[-1,-1,3]; [n le 3 select I[n] else 9*Self(n-1)-26*Self(n-2)+24*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 12 2014

A137786:=n->4^n - 3^n - 2^n; seq(A137786(n), n=0..25); # Wesley Ivan Hurt, Feb 10 2014

Table[4^n - 3^n - 2^n, {n, 0, 25}] (* Bruno Berselli, Jul 04 2012 *)
LinearRecurrence[{9,-26,24},{-1,-1,3},30] (* Harvey P. Dale, Sep 19 2012 *)
CoefficientList[Series[-(1 - 8 x + 14 x^2)/((1 - 2 x) (1 - 3 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)

a(n) = 4^n-3^n-2^n; \\ Joerg Arndt, Jul 04 2012

print([4**n - 3**n - 2**n for n in range(99)])
# Alex Ratushnyak, Jul 03 2012

G.f.: -(1-8*x+14*x^2)/((1-2*x)*(1-3*x)*(1-4*x)). - Bruno Berselli, Jul 04 2012
a(0)=-1, a(1)=-1, a(2)=3, a(n) = 9*a(n-1) - 26*a(n-2) + 24*a(n-3). - Harvey P. Dale, Sep 19 2012
E.g.f.: exp(2*x)*(exp(2*x) - exp(x) - 1). - Elmo R. Oliveira, Sep 12 2024

Offset set to 0, terms corrected, more terms added by Alex Ratushnyak, Jul 03 2012.

A231919 a(n) = 3^n + (4^n - 3^n) * (d(n) - 3), where d(n) = A000005(n).

Original entry on oeis.org

1, 2, -10, 81, -538, 4096, -12010, 65536, 19683, 1048576, -3840010, 49268766, -63920218, 268435456, 1073741824, 8546887871, -16921588858, 205383589230, -272553384010, 3291561314526, 4398046511104, 17592186044416, -70180457820010, 1406245165407356, 847288609443

Offset: 1

Wesley Ivan Hurt, Nov 15 2013

a(n) is negative if and only if n is an odd prime (A065091). If n is prime, then a(n) = - A002250(n). If n is a semiprime (A001358), a(n) gives the n-th power of the number of divisors of n. For example, a(4) = d(4)^4 = 3^4 = 81. Similarly, a(6) = d(6)^6 = 4^6 = 4096.

Cf. A000005, A002250.

with(numtheory); A231919:=n->3^n+(4^n-3^n)*(tau(n)-3); seq(A231919(n), n=1..100);

Table[3^n + (4^n - 3^n)(DivisorSigma[0,n] - 3), {n,100}]

a(n) = A000244(n) + A005061(n) * (A000005(n) - 3).

A264766 Irregular symmetric triangle of coefficients T(n,k) of the polynomials p(n,x) = Sum_{k=0..n} binomial(n+1,k)(1+x)^(2k)(-x)^(n-k) for 0 <= k <= 2n.

Original entry on oeis.org

1, 2, 3, 2, 3, 9, 13, 9, 3, 4, 18, 40, 51, 40, 18, 4, 5, 30, 90, 165, 201, 165, 90, 30, 5, 6, 45, 170, 405, 666, 783, 666, 405, 170, 45, 6, 7, 63, 287, 840, 1736, 2646, 3039, 2646, 1736, 840, 287, 63, 7, 8, 84, 448, 1554, 3864, 7224, 10424, 11763, 10424, 7224, 3864, 1554, 448, 84, 8, 9, 108, 660, 2646, 7686, 17010, 29520, 40851, 45481, 40851, 29520, 17010, 7686, 2646, 660, 108, 9, 10

Offset: 0

Werner Schulte, Nov 23 2015

			The irregular triangle T(n,k) begins:
n\k:  0   1    2     3     4     5      6      7      8     9    10  11  12
  0:  1
  1:  2   3    2
  2:  3   9   13     9     3
  3:  4  18   40    51    40    18      4
  4:  5  30   90   165   201   165     90     30      5
  5:  6  45  170   405   666   783    666    405    170    45     6
  6:  7  63  287   840  1736  2646   3039   2646   1736   840   287  63   7
  etc.
The polynomial corresponding to row 2 is p(2,x) = 3 + 9*x + 13*x^2 + 9*x^3 + 3*x^4.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000108, A000984, A001006, A002426, A003462, A005061, A027907, A058987, A163774, A260056.

T[n_, k_] := Sum[(-1)^j*Binomial[n + 1, j + 1]*Binomial[2*n - 2*j, k - j], {j, 0, n - Abs[k - n]}]; Table[T[n, k], {n,0,10}, {k,0,2*n}] // Flatten (* G. C. Greubel, Aug 12 2017 *)

T(n,k) = sum(j=0, n-abs(k-n), (-1)^j*binomial(n+1,j+1)*binomial(2*n-2*j,k-j));
tabf(nn) = for (n=0, nn, for (k=0, 2*n, print1(T(n, k), ", ");); print();); \\ Michel Marcus, Nov 24 2015

T(n,k) = Sum_{j=0..n-d} (-1)^j*binomial(n+1,j+1)*binomial(2*n-2*j,k-j) if d = 0 or better d = abs(k-n), and 0 <= k <= 2*n.
Recurrence: T(n,0) = n+1, and T(n,k) = 0 for k < 0 or k > 2*n, and T(n+1,k) = T(n,k-2) + T(n,k-1) + T(n,k) + binomial(2*n+2,k) for k > 0 and n >= 0.
T(n,k) = T(n,2*n-k) for 0 <= k <= 2*n.
p(n,x) = Sum_{k=0..2*n} T(n,k)*x^k = Sum_{k=0..n} (1+x)^(2*k)*(1+x+x^2)^(n-k) = Sum_{k=0..n} binomial(n+1,k)*(1+x+x^2)^k*x^(n-k) for n >= 0.
Recurrence: p(0,x) = 1, and p(n+1,x) = (1+x+x^2)*p(n,x)+(1+x)^(2*n+2), n >= 0.
T(n,n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n+1,j)*binomial(2*j,j) = A000984(n+1)-A002426(n+1) for n >= 0 (see also A163774).
Sum_{n>=0} T(n,n)*x^(n+1) = 1/sqrt(1-4*x) - 1/sqrt(1-2*x-3*x^2) for abs(x) < 1/4.
T(n,n-1) = binomial(2*n+2,n) - A027907(n+1,n) for n > 0.
T(n+1,n)/(n+2) = A000108(n+2) - A001006(n+1) for n >= 0 (see also A058987).
Row sums: p(n,1) = A005061(n+1) for n >= 0.
Alternating row sums: p(n,-1) = 1 for n >= 0.
p(n,-2) = Sum_{k=0..2*n} T(n,k)*(-2)^k = A003462(n+1) for n >= 0.
T(n,k) = Sum_{j=0..k} (-1)^j*A260056(n,j)*binomial(2*n-j,k-j) for 0 <= k <= 2*n.
A260056(n,k) = Sum_{j=0..k} (-1)^j*T(n,j)*binomial(2*n-j,k-j) for 0 <= k <= 2*n.
p(n,-1-x) = Sum{k=0..2*n} A260056(n,k)*x^(2*n-k) for n >= 0.
p(n,-x/(1+x))*(1+x)^(2*n) = Sum_{k=0..2*n} A260056(n,k)*x^k for n >= 0.
Sum_{n>=0} p(n,x)*t^n = 1/((1-t*(1+x)^2)*(1-t*(1+x+x^2))).
p(n,x)*x = (1+x)^(2*n+2) - (1+x+x^2)^(n+1), n >= 0.
T(n,k) = binomial(2*n+2,k+1) - A027907(n+1,k+1) for 0 <= k <= 2*n.

A359200 Triangle read by rows: T(n, k) = A358125(n,k)*binomial(n-1, k), 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 8, 3, 7, 30, 30, 7, 15, 88, 144, 88, 15, 31, 230, 520, 520, 230, 31, 63, 564, 1620, 2240, 1620, 564, 63, 127, 1330, 4620, 8120, 8120, 4620, 1330, 127, 255, 3056, 12432, 26432, 33600, 26432, 12432, 3056, 255, 511, 6894, 32112, 79968, 122976, 122976, 79968, 32112, 6894, 511

Offset: 1

Ambrosio Valencia-Romero, Dec 20 2022

			Triangle begins:
   0;
   1,     1;
   3,     8,     3;
   7,    30,    30,      7;
  15,    88,   144,     88,     15;
  31,   230,   520,    520,    230,     31;
  63,   564,  1620,   2240,   1620,    564,     63;
 127,  1330,  4620,   8120,   8120,   4620,   1330,    127;
 255,  3056, 12432,  26432,  33600,  26432,  12432,   3056,   255;
 511,  6894, 32112,  79968, 122976, 122976,  79968,  32112,  6894,   511;
1023, 15340, 80460, 229440, 413280, 499968, 413280, 229440, 80460, 15340, 1023;
...

Row sums give 2*A005061(n-1).

T := n -> local k; seq((2^n - 2^(n - k - 1) - 2^k)*binomial(n - 1, k), k = 0 .. n - 1);
seq(T(n), n = 1 .. 11);

T[n_, k_] := (2^n - 2^(n - k - 1) - 2^k)*Binomial[n - 1, k]; Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, Dec 20 2022 *)

T(n, k) = (2^n - 2^(n-k-1) - 2^k)*binomial(n-1, k), for n >= 1 and 0 <= k <= n-1.

A369892 Array read by antidiagonals: T(m, n) is the number of m X n binary arrays with a path of adjacent 1's from top row to bottom row using only left, right, and downward steps.

Original entry on oeis.org

1, 3, 1, 7, 7, 1, 15, 37, 17, 1, 31, 175, 197, 41, 1, 63, 781, 1985, 1041, 99, 1, 127, 3367, 18621, 22193, 5503, 239, 1, 255, 14197, 167337, 433801, 247759, 29089, 577, 1, 511, 58975, 1461797, 8057625, 10056087, 2764991, 153769, 1393, 1, 1023, 242461, 12519345, 144762849, 384409519, 232777209, 30856705, 812849, 3363, 1

Offset: 1

Caleb Stanford, Feb 05 2024

Similar to A359576 but disallowing Up steps.
The sequences are initially similar but differ for 4 X 5 grids (433801 instead of 433809), 4 X 6 grids (8057625 instead of 8057905), and 5 X 5 grids (10056087 instead of 10056959)
Can be calculated by dynamic programming from 1 X n grids to m X n grids by keeping track of the number of grids with each of the 2^n patterns of reachable squares in the last row.
Each row and each column satisfies a linear recurrence with constant coefficients. - Pontus von Brömssen, Feb 05 2025

			For the 37 2 X 3 grids, see A359576.
The following 4 X 5 grid is a counterexample that is counted by A359576 but not by the present sequence:
    10000
    10111
    11101
    00001
Notice that there is a path of 1s from the top to the bottom, but only via the upward step detour in the third column. There are 8 such 4 X 5 grids, formed from the above by reflection and by toggling the first row, second column and last row, second to last column.
Table starts:
    1      3        7         15          31          63         127 ...
    1      7       37        175         781        3367       14197 ...
    1     17      197       1985       18621      167337     1461797 ...
    1     41     1041      22193      433801     8057625   144762849 ...
    1     99     5503     247759    10056087   384409519   ...
    1    239    29089    2764991   232777209   ...
    1    577   153769   30856705   ...
    1   1393   812849   ...
    1   3363   ...
    1   ...
    ...

Caleb Stanford, Rust program to compute the sequence.

First 4 rows are A000225, A005061, A069361, A368809.
First 4 columns are A000012, A001333, A069378, A069379.
Cf. A359576 (up steps allowed).

cp's OEIS Frontend

A069383 Number of n X 8 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069384 Number of n X 9 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069385 Number of n X 10 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069386 Number of n X 11 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069387 Number of n X 12 binary arrays with a path of adjacent 1's from top row to bottom row.

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A137786 a(n) = 4^n - 3^n - 2^n.

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A231919 a(n) = 3^n + (4^n - 3^n) * (d(n) - 3), where d(n) = A000005(n).

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A264766 Irregular symmetric triangle of coefficients T(n,k) of the polynomials p(n,x) = Sum_{k=0..n} binomial(n+1,k)*(1+x)^(2*k)*(-x)^(n-k) for 0 <= k <= 2*n.

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A359200 Triangle read by rows: T(n, k) = A358125(n,k)*binomial(n-1, k), 0 <= k <= n-1.

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A264766 Irregular symmetric triangle of coefficients T(n,k) of the polynomials p(n,x) = Sum_{k=0..n} binomial(n+1,k)(1+x)^(2k)(-x)^(n-k) for 0 <= k <= 2n.