A065941 -id:A065941 - cp's OEIS frontend

A131774 2*A065941 - A049310.

1, 2, 1, 1, 2, 1, 2, 0, 4, 1, 1, 2, 3, 4, 1, 2, -1, 8, 2, 6, 1, 1, 2, 4, 8, 7, 6, 1, 2, -2, 12, 0, 20, 6, 8, 1, 1, 2, 4, 12, 15, 20, 13, 8, 1, 2, -3, 16, -6, 42, 9, 40, 12, 10, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = the Lucas numbers, A000032, starting (1, 3, 4, 7, 11, ...). Generally, N*A065941 - (N-1)*A049310 = triangles with row sums = Fibonacci-like sequences starting (1, (N+1), (N+1+1), ...). With N = 2, row sums of the triangle A131774 = (1, 3, 4, 7, ...).

			First few rows of the triangle:
  1;
  2,  1;
  1,  2,  1;
  2,  0,  4,  1;
  1,  2,  3,  4,  1;
  2, -1,  8,  2,  6,  1;
  1,  2,  4,  8,  7,  6,  1;
  ...

Cf. A065941, A049310, A000032, A131775, A131776, A131777, A131778.

2*A065941 - A049310 as infinite lower triangular matrices.

A065942 Central column of triangle A065941.

Original entry on oeis.org

1, 1, 3, 4, 15, 21, 84, 120, 495, 715, 3003, 4368, 18564, 27132, 116280, 170544, 735471, 1081575, 4686825, 6906900, 30045015, 44352165, 193536720, 286097760, 1251677700, 1852482996, 8122425444, 12033222880, 52860229080, 78378960360

Offset: 0

Len Smiley, Nov 29 2001

When viewed as (1,1), (3,4), (15,21), ... this represents a shallow staircase on Pascal's triangle, arranged as a square array. - Paul Barry, Mar 11 2003
Also central column of triangle A011973 (taking rows with odd number of terms only). - John Molokach, Jul 08 2013
Interleaving of A005809 and A045721. - Bruce J. Nicholson, Apr 24 2018

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 15*x^4 + 21*x^5 + 84*x^6 + 120*x^7 + ... - _Michael Somos_, Jun 23 2018

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)

Michael De Vlieger, Table of n, a(n) for n = 0..2416
Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271; Corrections.

Cf. A065941 (complete triangle), A047749.

Cf. A005809, A045721.

List([0..40],n->Binomial(n+Int(n/2),n)); # Muniru A Asiru, Apr 28 2018

Array[Binomial[# + Floor[#/2], #] &, 30, 0] (* Michael De Vlieger, Apr 27 2018 *)

a(n) = binomial(n+n\2, n); \\ Altug Alkan, Apr 24 2018

a(n) = binomial(2n-floor((n+1)/2), floor(n/2)).
a(n+1) = Sum_{k=0..ceiling(n/2)} binomial(n+k, k). - Benoit Cloitre, Mar 06 2004
a(n) = binomial(n+floor(n/2), n). - Paul Barry, May 18 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+k, k). - Paul Barry, Jul 06 2004
a(2n-1) = binomial(3n-3,n-1); a(2n) = binomial(3n-2,n-1). - John Molokach, Jul 08 2013
G.f.: A(x) = x*(d/dx)[log(S(x)-1)] = x*[(d/dx) S(x)]/[S(x)-1], where S(x) is the g.f. of A047749. - Vladimir Kruchinin, Jun 12 2014.
Conjecture: 8*n*(n-1)*a(n) -36*(n-1)*(n-3)*a(n-1) +6*(-9*n^2+18*n-14)*a(n-2) +27*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Jun 13 2014
0 = a(n)*(+281138850*a(n+2) +729089100*a(n+3) -77071527*a(n+4) -134472793*a(n+5)) +a(n+1)*(+15618825*a(n+2) -1650969*a(n+3) -9342280*a(n+4) -1729448*a(n+5)) +a(n+2)*(-19089675*a(n+2) -61394833*a(n+3) +6470716*a(n+4) +14929796*a(n+5)) +a(n+3)*(-1291668*a(n+3) +553572*a(n+4) +246032*a(n+5)) for all n in Z. - Michael Somos, Jun 23 2018

A131332 Triangle read by rows: A065941 * A097807.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2, -1, 2, 1, 1, 3, -2, 3, 1, 2, 1, 5, -4, 5, 0, 4, 2, 1, 8, -7, 8, -2, 7, 3, 3, 1, 13, -12, 13, -6, 12, 3, 7, 3, 1, 21, -20, 21, -13, 20, 1, 14, 6, 4, 1, 34, -33, 34, -25, 33, -5, 26, 9, 11, 4, 1, 55, -54, 55, -45, 54, -18, 46, 10, 25, 10, 5, 1, 89, -88, 89, -78, 88, -43, 79, 5, 51, 19, 16, 5, 1

Offset: 1

Gary W. Adamson, Jun 29 2007

Row sums = Fibonacci numbers: (1, 1, 2, 3, 5, 8, ...).

			First few rows of the triangle are:
  1;
  0,  1;
  1,  0, 1;
  1,  0, 1,  1;
  2, -1, 2,  1, 1;
  3, -2, 3,  1, 2, 1;
  5, -4, 5,  0, 4, 2, 1;
  8, -7, 8, -2, 7, 3, 3, 1;
  ...

Cf. A097807, A131333, A131334, A131335, A131336.

A065941 * A000012(signed by columns, + - + - ...).

a(47) = -20 corrected, name edited and more terms from Georg Fischer, Jun 05 2023

A131334 A000012(signed) * A065941.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 2, 2, 2, 1, 1, 0, 3, 2, 4, 2, 1, 0, 1, 3, 3, 6, 4, 3, 1, 1, 0, 4, 3, 9, 6, 7, 3, 1, 0, 1, 4, 4, 12, 9, 13, 7, 4, 1

Offset: 1

Gary W. Adamson, Jun 29 2007

Row sums = Fibonacci numbers.

			First few rows of the triangle are:
1;
0, 1;
1, 0, 1;
0, 1, 1, 1;
1, 0, 2, 1, 1;
0, 1, 2, 2, 2, 1;
1, 0, 3, 2, 4, 2, 1;
0, 1, 3, 3, 6, 4, 3, 1;
1, 0, 4, 3, 9, 6, 7, 3, 1;
...

Cf. A131332, A131333, A131335, A131336.

A000012(signed by columns, + - + -...) * A065941.

A131775 3A065941 - 2A049310.

Original entry on oeis.org

1, 3, 1, 1, 3, 1, 3, -1, 6, 1, 1, 3, 3, 6, 1, 3, -3, 12, 1, 9, 1, 1, 3, 3, 12, 8, 9, 1, 3, -5, 18, -5, 30, 6, 12, 1, 1, 3, 1, 18, 15, 30, 16, 12, 1, 3, -7, 24, -19, 63, 3, 60, 14, 15, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = A000285, (a Fibonacci-like sequence) starting (1, 4, 5, 9, 14, 23, ...).

			Table begins:
  1;
  3,  1;
  1,  3,  1;
  3, -1,  6,  1;
  1,  3,  3,  6,  1;
  3, -3, 12,  1,  9,  1;
  1,  3,  3, 12,  8,  9,  1;
  3, -5, 18, -5, 30,  6, 12,  1;
  1,  3,  1, 18, 15, 30, 16, 12,  1;
  ...

Cf. A065941, A049310, A131774, A131776, A131777, A131778.

3*A065941 - 2*A049310.

A131776 4A065941 - 3A049310.

Original entry on oeis.org

1, 4, 1, 1, 4, 1, 4, -2, 8, 1, 1, 4, 3, 8, 1, 4, -5, 16, 0, 12, 1, 1, 4, 2, 16, 9, 12, 1, 4, -8, 24, -10, 40, 6, 16, 1, 1, 4, -2, 24, 15, 40, 19, 16, 1, 4, -11, 32, -32, 84, -3, 80, 16, 20, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = A022095, a Fibonacci-like sequence starting (1, 5, 6, 11, 17, 28, ...).

			First few rows of the triangle:
  1;
  4,  1;
  1,  4,  1;
  4, -2,  8,  1;
  1,  4,  3,  8,  1;
  4, -5, 16,  0, 12,  1;
  1,  4,  2, 16,  9, 12,  1;
  ...

Cf. A065941, A049310, A022095, A131774, A131775, A131777, A131778.

4*A065941 - 3*A049310 as infinite lower triangular matrices.

A131243 A065941^2 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 4, 1, 8, 7, 10, 4, 1, 13, 12, 23, 12, 6, 1, 21, 20, 48, 29, 21, 6, 1, 34, 33, 96, 64, 62, 24, 8, 1, 55, 54, 185, 132, 160, 74, 36, 8, 1, 89, 88, 348, 261, 382, 200, 130, 40, 10, 1, 144, 143, 642, 500, 859, 492, 400, 150, 55, 10, 1

Offset: 0

Gary W. Adamson, Jun 22 2007

Left border, Fibonacci numbers; next border to the right, (Fibonacci numbers - 1).

			First few rows of the triangle:
   1;
   2,  1;
   3,  2,   1;
   5,  4,   4,   1;
   8,  7,  10,   4,   1;
  13, 12,  23,  12,   6,   1;
  21, 20,  48,  29,  21,   6,   1;
  34, 33,  96,  64,  62,  24,   8,  1;
  55, 54, 185, 132, 160,  74,  36,  8,  1;
  89, 88, 348, 261, 382, 200, 130, 40, 10, 1;

Hakan Icoz, Rows 0..100, flattened

Cf. A065941, A131244 (row sums), A131245, A131246.

Data and Example corrected by Jon E. Schoenfield, Feb 26 2022

More terms from Hakan Icoz, Jan 23 2023

A131777 5A065941 - 4A049310.

Original entry on oeis.org

1, 5, 1, 1, 5, 1, 5, -3, 10, 1, 1, 5, 3, 10, 1, 5, -7, 20, -1, 15, 1, 1, 5, 1, 20, 10, 15, 1, 5, -11, 30, -15, 50, 6, 20, 1, 1, 5, -5, 30, 15, 50, 22, 20, 1, 5, -15, 40, -45, 105, -9, 100, 18, 25, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = A022096(n-1).

			First few rows of the triangle are:
1;
5, 1
1, 5, 1;
5, -3, 10, 1;
1, 5, 3, 10, 1;
5, -7, 20, -1, 15, 1;
1, 5, 1, 20, 10, 15, 1;
...

Cf. A131774, A131775, A131776, A065941, A049310, A022096.

5*A065941 - 4*A049310 as infinite lower triangular matrices.

Corrected A-number in row sums reference R. J. Mathar, Jun 16 2009

A131778 6A065941 - 5A049310.

Original entry on oeis.org

1, 6, 1, 1, 6, 1, 6, -4, 12, 1, 1, 6, 3, 12, 1, 6, -9, 24, -2, 18, 1, 1, 6, 0, 24, 11, 18, 1, 6, -14, 36, -20, 60, 6, 24, 1, 1, 6, -8, 36, 15, 60, 25, 24, 1, 6, -19, 48, -58, 126, -15, 120, 20, 30, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = A022097, a Fibonacci-like sequence starting (1, 7, 8, 15, 23, 38, ...).

			First few rows of the triangle:
  1;
  6,  1;
  1,  6,  1;
  6, -4, 12,  1;
  1,  6,  3, 12,  1;
  6, -9, 24, -2, 18,  1;
  1,  6,  0, 24, 11, 18,  1;
  ...

Cf. A065941, A049310, A099097, A131774, A131775, A131776, A131777.

6*A065941 - 5*A049310 as infinite lower triangular matrices.

A131402 2*A007318 - (A046854 + A065941 - A000012).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 7, 6, 1, 1, 7, 14, 14, 7, 1, 1, 9, 20, 33, 20, 9, 1, 1, 10, 31, 56, 56, 31, 10, 1, 1, 12, 40, 97, 111, 97, 40, 12, 1, 1, 13, 55, 142, 217, 217, 142, 55, 13, 1, 1, 15, 67, 213, 358, 463, 358, 213, 67, 15, 1, 1, 16, 86, 287, 590, 841, 841, 590, 287, 86, 16, 1

Offset: 0

Gary W. Adamson, Jul 07 2007

Row sums = A131403: (1, 2, 5, 10, 21, 44, 93, ...).

			First few rows of the triangle are:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  4,  1;
  1,  6,  7,  6,  1;
  1,  7, 14, 14,  7,  1;
  1,  9, 20, 33, 20,  9,  1;
  1, 10, 31, 56, 56, 31, 10,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A131403.

Cf. A046854, A065941.

T(n,k) = if(k <= n, 2*binomial(n, k) + 1 - binomial((n + k)\2, k) - binomial(n-(k+1)\2, k\2), 0) \\ Andrew Howroyd, Aug 09 2018

2*A007318 - (A046854 + A065941 - A000012) as infinite lower triangular matrices.

T(n,k) = 2*binomial(n, k) + 1 - binomial(floor((n + k)/2), k) - binomial(n-floor((k+1)/2), floor(k/2)). - Andrew Howroyd, Aug 09 2018

Missing terms inserted and a(55) and beyond from Andrew Howroyd, Aug 09 2018

cp's OEIS Frontend

A131774 2*A065941 - A049310.

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A065942 Central column of triangle A065941.

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A131332 Triangle read by rows: A065941 * A097807.

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A131334 A000012(signed) * A065941.

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A131775 3*A065941 - 2*A049310.

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A131776 4*A065941 - 3*A049310.

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A131243 A065941^2 as an infinite lower triangular matrix.

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A131777 5*A065941 - 4*A049310.

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A131778 6*A065941 - 5*A049310.

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A131775 3A065941 - 2A049310.

A131776 4A065941 - 3A049310.

A131777 5A065941 - 4A049310.

A131778 6A065941 - 5A049310.