A131066 -id:A131066 - cp's OEIS frontend

A131061 Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.

1, 1, 1, 1, 5, 1, 1, 9, 9, 1, 1, 13, 21, 13, 1, 1, 17, 37, 37, 17, 1, 1, 21, 57, 77, 57, 21, 1, 1, 25, 81, 137, 137, 81, 25, 1, 1, 29, 109, 221, 277, 221, 109, 29, 1, 1, 33, 141, 333, 501, 501, 333, 141, 33, 1, 1, 37, 177, 477, 837, 1005, 837, 477, 177, 37, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A131062: (1, 2, 7, 20, 49, 110, 235, ...); the binomial transform of (1, 1, 4, 4, 4, ...).

Triangle equals 4*A007318 - 3*A000012 as infinite lower triangular matrices. - Emeric Deutsch, Jun 21 2007

			First few rows of the triangle are
  1;
  1,  1;
  1,  5,  1;
  1,  9,  9,  1;
  1, 13, 21, 13,  1;
  1, 17, 37, 37, 17,  1;
  1, 21, 57, 77, 57, 21, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A109128, A123203, A131060, A131063, A131064, A131065, A131066, A131067, A131068.

[4*Binomial(n, k) -3: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

T := proc (n, k) if k <= n then 4*binomial(n, k)-3 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 21 2007

Table[4*Binomial[n, k] -3, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[4*binomial(n, k) -3 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

G.f.:(1 - z - t*z + 4*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 21 2007

More terms from Emeric Deutsch, Jun 21 2007

A131065 Triangle read by rows: T(n,k) = 6*binomial(n,k) - 5 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 31, 19, 1, 1, 25, 55, 55, 25, 1, 1, 31, 85, 115, 85, 31, 1, 1, 37, 121, 205, 205, 121, 37, 1, 1, 43, 163, 331, 415, 331, 163, 43, 1, 1, 49, 211, 499, 751, 751, 499, 211, 49, 1, 1, 55, 265, 715, 1255, 1507, 1255, 715, 265, 55, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A131066.
The matrix inverse starts:
       1;
      -1,       1;
       6,      -7,      1;
     -66,      78,    -13,      1;
    1086,   -1284,    216,    -19,   1;
  -23826,   28170,  -4740,    420, -25,   1;
  653406, -772536, 129990, -11520, 690, -31, 1; - R. J. Mathar, Mar 12 2013

			First few rows of the triangle are:
  1;
  1,  1;
  1,  7,  1;
  1, 13, 13,  1;
  1, 19, 31, 19,  1;
  1, 25, 55, 55, 25, 1;
...

Indranil Ghosh, Rows 0..120 of triangle, flattened

Cf. A109128, A123203, A131060, A131061, A131063, A131064, A131066, A131067, A131068.

[6*Binomial(n,k) -5: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

T := proc (n, k) if k <= n then 6*binomial(n, k)-5 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007

Table[6*Binomial[n,k]-5,{n,0,15},{k,0,n}]//Flatten (* Harvey P. Dale, May 15 2016 *)

[[6*binomial(n,k) -5 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

G.f.: (1-z-t*z+6*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

More terms from Emeric Deutsch, Jun 20 2007

A131060 3A007318 - 2A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 16, 10, 1, 1, 13, 28, 28, 13, 1, 1, 16, 43, 58, 43, 16, 1, 1, 19, 61, 103, 103, 61, 19, 1, 1, 22, 82, 166, 208, 166, 82, 22, 1, 1, 25, 106, 250, 376, 376, 250, 106, 25, 1, 1, 28, 133, 358, 628, 754, 628, 358, 133, 28, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A097813: (1, 2, 6, 16, 38, 84, 178, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  4,  1;
  1,  7,  7,  1;
  1, 10, 16, 10,  1;
  1, 13, 28, 28, 13,  1;
  1, 16, 43, 58, 43, 16,  1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A109128, A123203, A131061, A131063, A131064, A131065, A131066, A131067, A131068.

[3*Binomial(n,k) -2: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

A131060:= (n,k) -> 3*binomial(n, k)-2; seq(seq(A131060(n, k), k = 0..n), n = 0.. 10); # G. C. Greubel, Mar 12 2020

T[n_, k_] = 3*Binomial[n, k] -2; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 20 2008 *)

[[3*binomial(n,k) -2 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

T(n,k) = 3*binomial(n,k) - 2. - Roger L. Bagula, Aug 20 2008

More terms from Roger L. Bagula, Aug 20 2008

A131063 Triangle read by rows: T(n,k) = 5*binomial(n,k) - 4 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 16, 26, 16, 1, 1, 21, 46, 46, 21, 1, 1, 26, 71, 96, 71, 26, 1, 1, 31, 101, 171, 171, 101, 31, 1, 1, 36, 136, 276, 346, 276, 136, 36, 1, 1, 41, 176, 416, 626, 626, 416, 176, 41, 1, 1, 46, 221, 596, 1046, 1256, 1046, 596, 221, 46, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A131064: (1, 2, 8, 24, 60, 136, 292, ...), the binomial transform of (1, 1, 5, 5, 5, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  6,  1;
  1, 11, 11,  1;
  1, 16, 26, 16,  1;
  1, 21, 46, 46, 21,  1;
  1, 26, 71, 96, 71, 26,  1;
  ...

Muniru A Asiru, Rows n=0..100 of triangle, flattened

Cf. A109128, A123203, A131060, A131061, A131064, A131065, A131066, A131067, A131068.

Print(Flat(List([0..10],n->List([0..n],k->5*Binomial(n,k)-4)))); # Muniru A Asiru, Feb 21 2019

[5*Binomial(n, k) -4: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

T := proc (n, k) if k <= n then 5*binomial(n, k)-4 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007

Table[5*Binomial[n,k] -4, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[5*binomial(n, k) -4 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

G.f.: (1-z-t*z+5*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

More terms from Emeric Deutsch, Jun 20 2007

A131067 Triangle read by rows: T(n,k) = 7*binomial(n,k) - 6 for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 15, 15, 1, 1, 22, 36, 22, 1, 1, 29, 64, 64, 29, 1, 1, 36, 99, 134, 99, 36, 1, 1, 43, 141, 239, 239, 141, 43, 1, 1, 50, 190, 386, 484, 386, 190, 50, 1, 1, 57, 246, 582, 876, 876, 582, 246, 57, 1, 1, 64, 309, 834, 1464, 1758, 1464, 834, 309, 64, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A131068: (1, 2, 10, 32, 82, 188, 406, ...), the binomial transform of (1, 1, 7, 7, 7, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  8,  1;
  1, 15, 15,  1;
  1, 22, 36, 22,  1;
  1, 29, 64, 64, 29, 1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A131064, A131066.

Sequence m*binomial(n,k) - (m-1): A007318 (m=1), A109128 (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), this sequence (m=7), A131068 (m=8).

[7*Binomial(n, k) -6: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

T := proc (n, k) if k <= n then 7*binomial(n, k)-6 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007

Table[7*Binomial[n, k] -6, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[7*binomial(n, k) -6 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

G.f.: G(t,z) = (1-z-t*z+7*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

More terms from Emeric Deutsch, Jun 20 2007

A123203 a(n) = 2^(n+1) - 3*n.

Original entry on oeis.org

1, 2, 7, 20, 49, 110, 235, 488, 997, 2018, 4063, 8156, 16345, 32726, 65491, 131024, 262093, 524234, 1048519, 2097092, 4194241, 8388542, 16777147, 33554360, 67108789, 134217650, 268435375, 536870828, 1073741737, 2147483558

Offset: 1

Gary W. Adamson, Jun 13 2007

An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 186, leads to this sequence. For the central square this vector leads to the companion sequence A036563. - Johannes W. Meijer, Aug 15 2010

			a(4) = 20, row sums of 4th row of triangle A131062: (1, 9, 9, 1).
a(4) = 20 = (1, 3, 3, 1) dot (1, 1, 4, 4) = (1 + 3 + 12 + 4).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Joseph Breen and Emma Copeland, Non-orientable Nurikabe, arXiv:2506.12612 [math.CO], 2025. See pp. 1, 4.
Tamas Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015) 73-94.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000079, A000295, A109128, A131060, A131061, A131063, A131064, A131065, A131066.

[2^(n+1) -3*n: n in [1..40]]; // G. C. Greubel, Sep 14 2024

Table[2^(n+1) - 3*n, {n,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
LinearRecurrence[{4,-5,2},{1,2,7},40] (* Harvey P. Dale, Mar 30 2024 *)

def A123203(n): return 2^(n+1) -3*n
[A123203(n) for n in range(1,41)] # G. C. Greubel, Sep 14 2024

Binomial transform of [1, 1, 4, 4, 4, ...].
Equals row sums of triangle A131061.
From Johannes W. Meijer, Aug 15 2010; corrected by Colin Barker, Jul 28 2012: (Start)
a(n) = 2^(1+n) - 3*n.
a(n) = 3*A000295(n-1) + A000079(n-1).
(End)
G.f.: x*(1 - 2*x + 4*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Jul 28 2012
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Colin Barker, Jul 29 2012
E.g.f.: 2*exp(2*x) - 3*x*exp(x) - 2. - G. C. Greubel, Sep 14 2024

More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008

Title changed by G. C. Greubel, Sep 14 2024

A131064 Binomial transform of [1, 1, 5, 5, 5, ...].

Original entry on oeis.org

1, 2, 8, 24, 60, 136, 292, 608, 1244, 2520, 5076, 10192, 20428, 40904, 81860, 163776, 327612, 655288, 1310644, 2621360, 5242796, 10485672, 20971428, 41942944, 83885980, 167772056, 335544212, 671088528, 1342177164, 2684354440

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums of triangle A131063. - Emeric Deutsch, Jun 20 2007

			a(3) = 24 = sum of row 4 terms of A131063: (1 + 11 + 11 + 1).
a(3) = 24 = (1, 3, 3, 1) dot (1, 1, 5, 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A109128, A123203, A131060, A131061, A131063, A131065, A131066, A131067, A131068.

Print(List([0..30],n->5*2^n-4*n-4)); # Muniru A Asiru, Feb 21 2019

I:=[1, 2, 8]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2) + 2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 05 2012

a := proc (n) options operator, arrow; 5*2^n-4*n-4 end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Jun 20 2007

CoefficientList[Series[(1-2x+5x^2)/((1-2x)(1-x)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 05 2012 *)
LinearRecurrence[{4,-5,2},{1,2,8},30] (* Harvey P. Dale, Dec 29 2014 *)

[5*2^n -4*(n+1) for n in (0..30)] # G. C. Greubel, Mar 12 2020

From Emeric Deutsch, Jun 20 2007: (Start)
a(n) = 5*2^n - 4*(n + 1).
G.f.: (1-2*x+5*x^2)/((1-2*x)*(1-x)^2). (End)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Vincenzo Librandi, Jul 05 2012
E.g.f.: 5*exp(2*x) - 4*(1+x)*exp(x). - G. C. Greubel, Mar 12 2020

Corrected and extended by Emeric Deutsch, Jun 20 2007

A131068 Binomial transform of [1, 1, 7, 7, 7, ...].

Original entry on oeis.org

1, 2, 10, 32, 82, 188, 406, 848, 1738, 3524, 7102, 14264, 28594, 57260, 114598, 229280, 458650, 917396, 1834894, 3669896, 7339906, 14679932, 29359990, 58720112, 117440362, 234880868, 469761886, 939523928, 1879048018, 3758096204

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums of triangle A131067. - Emeric Deutsch, Jun 20 2007

			a(3) = 32 = sum of row 4 terms of triangle A131067: (1 + 15 + 15 + 1).
a(3) = 32 = (1, 3, 3, 1) dot (1, 1, 7, 7) = (1 + 3 + 21 + 7).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A131060, A131061, A131063, A131064, A131065, A131066, A131067.

I:=[1, 2, 10]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2) + 2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 05 2012

a := proc (n) options operator, arrow; 7*2^n-6*n-6 end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Jun 20 2007

Table[7*2^n -6*(n+1), {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2008; modified by G. C. Greubel, Mar 12 2020 *)
CoefficientList[Series[(1-2x+7x^2)/((1-2x)(1-x)^2),{x,0,40}],x] (* Vincenzo Librandi, Jul 05 2012 *)

vector(31, n, my(m=n-1); 7*2^m -6*(m+1) ) \\ G. C. Greubel, Mar 12 2020

[7*2^n -6*(n+1) for n in (0..30)] # G. C. Greubel, Mar 12 2020

From Emeric Deutsch, Jun 20 2007: (Start)
a(n) = 7*2^n - 6*(n + 1).
G.f.: (1 - 2*x + 7*x^2)/((1-2*x)*(1-x)^2). (End)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Vincenzo Librandi, Jul 05 2012
E.g.f.: 7*exp(2*x) - 6*(1+x)*exp(x). - G. C. Greubel, Mar 12 2020

Corrected and extended by Emeric Deutsch, Jun 20 2007

cp's OEIS Frontend

A131061 Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.

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A131065 Triangle read by rows: T(n,k) = 6*binomial(n,k) - 5 for 0 <= k <= n.

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A131060 3*A007318 - 2*A000012 as infinite lower triangular matrices.

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A131063 Triangle read by rows: T(n,k) = 5*binomial(n,k) - 4 for 0 <= k <= n.

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A131067 Triangle read by rows: T(n,k) = 7*binomial(n,k) - 6 for 0 <= k <= n.

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A123203 a(n) = 2^(n+1) - 3*n.

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A131060 3A007318 - 2A000012 as infinite lower triangular matrices.