A194005 -id:A194005 - cp's OEIS frontend

A085478 Triangle read by rows: T(n, k) = binomial(n + k, 2*k).

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 15, 7, 1, 1, 15, 35, 28, 9, 1, 1, 21, 70, 84, 45, 11, 1, 1, 28, 126, 210, 165, 66, 13, 1, 1, 36, 210, 462, 495, 286, 91, 15, 1, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 1, 55, 495, 1716, 3003, 3003, 1820, 680, 153, 19, 1

Offset: 0

Philippe Deléham, Aug 14 2003

Coefficient array for Morgan-Voyce polynomial b(n,x). A053122 (unsigned) is the coefficient array for B(n,x). Reversal of A054142. - Paul Barry, Jan 19 2004
This triangle is formed from even-numbered rows of triangle A011973 read in reverse order. - Philippe Deléham, Feb 16 2004
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k+1 peaks. T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k peaks at height >= 2. T(n,k) is the number of directed column-convex polyominoes of area n+1, having k+1 columns. - Emeric Deutsch, May 31 2004
Riordan array (1/(1-x), x/(1-x)^2). - Paul Barry, May 09 2005
The triangular matrix a(n,k) = (-1)^(n+k)*T(n,k) is the matrix inverse of A039599. - Philippe Deléham, May 26 2005
The n-th row gives absolute values of coefficients of reciprocal of g.f. of bottom-line of n-wave sequence. - Floor van Lamoen (fvlamoen(AT)planet.nl), Sep 24 2006
Unsigned version of A129818. - Philippe Deléham, Oct 25 2007
T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k >=1 (height(alpha) = |Im(alpha)|) and of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Oct 02 2008
A085478 is jointly generated with A078812 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n>1, u(n,x) = u(n-1,x)+x*v(n-1)x and v(n,x) = u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 25 2012
Per Kimberling's recursion relations, see A102426. - Tom Copeland, Jan 19 2016
Subtriangle of the triangle given by (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012
T(n,k) is also the number of compositions (ordered partitions) of 2*n+1 into 2*k+1 parts which are all odd. Proof: The o.g.f. of column k, x^k/(1-x)^(2*k+1) for k >= 0, is the o.g.f. of the odd-indexed members of the sequence with o.g.f. (x/(1-x^2))^(2*k+1) (bisection, odd part). Thus T(n,k) is obtained from the sum of the multinomial numbers A048996 for the partitions of 2*n+1 into 2*k+1 parts, all of which are odd. E.g., T(3,1) = 3 + 3 from the numbers for the partitions [1,1,5] and [1,3,3], namely 3!/(2!*1!) and 3!/(1!*2!), respectively. The number triangle with the number of these partitions as entries is A152157. - Wolfdieter Lang, Jul 09 2012
The matrix elements of the inverse are T^(-1)(n,k) = (-1)^(n+k)*A039599(n,k). - R. J. Mathar, Mar 12 2013
T(n,k) = A258993(n+1,k) for k = 0..n-1. - Reinhard Zumkeller, Jun 22 2015
The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the algebraic function F(x)*G(x)^n about 0, where F(x) = (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) and G(x) = ((1 + sqrt(1 + 4*x))/2)^2. For example, for n = 4, (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) * ((1 + sqrt(1 + 4*x))/2)^8 = (x^4  + 10*x^3 + 15*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 23 2018
Row n also gives the coefficients of the characteristc polynomial of the tridiagonal n X n matrix M_n given in A332602: Phi(n, x) := Det(M_n - x*1_n) = Sum_{k=0..n} T(n, k)*(-x)^k, for n >= 0, with Phi(0, x) := 1. - Wolfdieter Lang, Mar 25 2020
It appears that the largest root of the n-th degree polynomial is equal to the sum of the distinct diagonals of a (2*n+1)-gon including the edge, 1. The largest root of x^3 - 6*x^2 + 5*x - 1 is 5.048917... = the sum of (1 + 1.80193... + 2.24697...). Alternatively, the largest root of the n-th degree polynomial is equal to the square of sigma(2*n+1). Check: 5.048917... is the square of sigma(7), 2.24697.... Given N = 2*n+1, sigma(N) (N odd) can be defined as 1/(2*sin(Pi/(2*N))). Relating to the 9-gon, the largest root of x^4 - 10*x^3 + 15*x^2 - 7*x + 1 is 8.290859..., = the sum of (1 + 1.879385... + 2.532088... + 2.879385...), and is the square of sigma(9), 2.879385... Refer to A231187 for a further clarification of sigma(7). - Gary W. Adamson, Jun 28 2022
For n >=1, the n-th row is given by the coefficients of the minimal polynomial of -4*sin(Pi/(4*n + 2))^2. - Eric W. Weisstein, Jul 12 2023
Denoting this lower triangular array by L, then L * diag(binomial(2*k,k)^2) * transpose(L) is the LDU factorization of A143007, the square array of crystal ball sequences for the A_n X A_n lattices. - Peter Bala, Feb 06 2024
T(n, k) is the number of occurrences of the periodic substring (01)^k in the periodic string (01)^n (see Proposition 4.7 at page 7 in Fang). - Stefano Spezia, Jun 09 2024

			Triangle begins as:
  1;
  1    1;
  1    3    1;
  1    6    5    1;
  1   10   15    7    1;
  1   15   35   28    9    1;
  1   21   70   84   45   11    1;
  1   28  126  210  165   66   13    1;
  1   36  210  462  495  286   91   15    1;
  1   45  330  924 1287 1001  455  120   17    1;
  1   55  495 1716 3003 3003 1820  680  153   19    1;
...
From _Philippe Deléham_, Mar 26 2012: (Start)
(0, 1, 0, 1, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, ...) begins:
  1
  0, 1
  0, 1,  1
  0, 1,  3,   1
  0, 1,  6,   5,   1
  0, 1, 10,  15,   7,   1
  0, 1, 15,  35,  28,   9,  1
  0, 1, 21,  70,  84,  45, 11,  1
  0, 1, 28, 126, 210, 165, 66, 13, 1. (End)

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - _N. J. A. Sloane_, May 10 2012
Peter Bala, A 4-parameter family of embedded Riordan arrays
E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.
Paul Barry and Aoife Hennessy, Notes on a Family of Riordan Arrays and Associated Integer Hankel Transforms , JIS 12 (2009) 09.5.3.
Eduardo H. M. Brietzke, Recurrence Relations for Sums of Binomial Coefficients and Some Generalizations, Journal of Integer Sequences, Vol. 27 (2024), Article 24.3.4.
E. Czabarka et al, Enumerations of peaks and valleys on non-decreasing Dyck paths, Disc. Math. 341 (2018) 2789-2807, Theorem 3.
Emeric Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Wenjie Fang, Maximal number of subword occurrences in a word, arXiv:2406.02971 [math.CO], 2024.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62. - _Abdullahi Umar_, Oct 02 2008
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174.
Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, (2005) 359-370.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Row sums: A001519. Signed versions: A123970, A129818.
Cf. A000108, A001006, A007318, A098158, A183160, A078812, A091042.
Cf. A258993, A025581, A051162, A054142, A332602.
Cf. A231187, A143007.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n+k, 2*k) ))); # G. C. Greubel, Aug 01 2019

a085478 n k = a085478_tabl !! n !! k
a085478_row n = a085478_tabl !! n
a085478_tabl = zipWith (zipWith a007318) a051162_tabl a025581_tabl
-- Reinhard Zumkeller, Jun 22 2015

[Binomial(n+k, 2*k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

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

(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A085478 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A078812 *) (*Clark Kimberling, Feb 25 2012 *)
(* Second program *)
Table[Binomial[n + k, 2 k], {n, 0, 12}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 01 2019 *)
CoefficientList[Table[Fibonacci[2 n + 1, Sqrt[x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Jul 03 2023 *)
Join[{{1}}, CoefficientList[Table[MinimalPolynomial[-4 Sin[Pi/(4 n + 2)]^2, x], {n, 20}], x]] (* Eric W. Weisstein, Jul 12 2023 *)

PARI
```
T(n,k) = binomial(n+k,n-k)
```

[[binomial(n+k,2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

T(n, k) = (n+k)!/((n-k)!*(2*k)!).
G.f.: (1-z)/((1-z)^2-tz). - Emeric Deutsch, May 31 2004
Row sums are A001519 (Fibonacci(2n+1)). Diagonal sums are A011782. Binomial transform of A026729 (product of lower triangular matrices). - Paul Barry, Jun 21 2004
T(n, 0) = 1, T(n, k) = 0 if n=0} T(n-1-j, k-1)*(j+1). T(0, 0) = 1, T(0, k) = 0 if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j). For the column k, g.f.: Sum_{n>=0} T(n, k)*x^n = (x^k) / (1-x)^(2*k+1). - Philippe Deléham, Feb 15 2004
Sum_{k=0..n} T(n,k)*x^(2*k) = A000012(n), A001519(n+1), A001653(n), A078922(n+1), A007805(n), A097835(n), A097315(n), A097838(n), A078988(n), A097841(n), A097727(n), A097843(n), A097730(n), A098244(n), A097733(n), A098247(n), A097736(n), A098250(n), A097739(n), A098253(n), A097742(n), A098256(n), A097767(n), A098259(n), A097770(n), A098262(n), A097773(n), A098292(n), A097776(n) for x=0,1,2,...,27,28 respectively. - Philippe Deléham, Dec 31 2007
T(2*n,n) = A005809(n). - Philippe Deléham, Sep 17 2009
A183160(n) = Sum_{k=0..n} T(n,k)*T(n,n-k). - Paul D. Hanna, Dec 27 2010
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Feb 06 2012
O.g.f. for column k: x^k/(1-x)^(2*k+1), k >= 0. [See the o.g.f. of the triangle above, and a comment on compositions. - Wolfdieter Lang, Jul 09 2012]
E.g.f.: (2/sqrt(x + 4))*sinh((1/2)*t*sqrt(x + 4))*cosh((1/2)*t*sqrt(x)) = t + (1 + x)*t^3/3! + (1 + 3*x + x^2)*t^5/5! + (1 + 6*x + 5*x^2 + x^3)*t^7/7! + .... Cf. A091042. - Peter Bala, Jul 29 2013
T(n, k) = A065941(n+3*k, 4*k) = A108299(n+3*k, 4*k) = A194005(n+3*k, 4*k). - Johannes W. Meijer, Sep 05 2013
Sum_{k=0..n} (-1)^k*T(n,k)*A000108(k) = A000007(n) for n >= 0. - Werner Schulte, Jul 12 2017
Sum_{k=0..floor(n/2)} T(n-k,k)*A000108(k) = A001006(n) for n >= 0. - Werner Schulte, Jul 12 2017
From  Peter Bala, Jun 26 2025: (Start)
The n-th row polynomial b(n, x) = (-1)^n * U(2*n, (i/2)*sqrt(x)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.
b(n, x) = (-1)^n * Dir(n, -1 - x/2), where Dir(n, x) is the n-th row polynomial of the triangle A244419.
b(n, -1 - x) is the n-th row polynomial of A098493. (End)

A034943 Binomial transform of Padovan sequence A000931.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217, 20330163, 47261895, 109870576, 255418101, 593775046, 1380359512, 3208946545

Offset: 0

N. J. A. Sloane

Trisection of the Padovan sequence: a(n) = A000931(3n). - Paul Barry, Jul 06 2004
a(n+1) gives diagonal sums of Riordan array (1/(1-x),x/(1-x)^3). - Paul Barry, Oct 11 2005
a(n+2) is the sum, over all Boolean n-strings, of the product of the lengths of the runs of 1. For example, the Boolean 7-string (0,1,1,0,1,1,1) has two runs of 1s. Their lengths, 2 and 3, contribute a product of 6 to a(9). The 8 Boolean 3-strings contribute to a(5) as follows: 000 (empty product), 001, 010, 100, 101 all contribute 1, 011 and 110 contribute 2, 111 contributes 3. - David Callan, Nov 29 2007
[a(n), a(n+1), a(n+2)], n > 0, = [0,1,0; 0,0,1; 1,-2,3]^n * [1,1,1]. - Gary W. Adamson, Mar 27 2008
Without the initial 1 and 1: 1, 2, 5, 12, 28, this is also the transform of 1 by the T_{1,0} transformation; see Choulet link. - Richard Choulet, Apr 11 2009
Without the first 1: transform of 1 by T_{0,0} transformation (see Choulet link). - Richard Choulet, Apr 11 2009
Starting (1, 2, 5, 12, ...) = INVERT transform of (1, 1, 2, 3, 4, 5, ...) and row sums of triangle A159974. - Gary W. Adamson, Apr 28 2009
a(n+1) is also the number of 321-avoiding separable permutations. (A permutation is separable if it avoids both 2413 and 3142.) - Vince Vatter, Sep 21 2009
a(n+1) is an eigensequence of the sequence array for (1,1,2,3,4,5,...). - Paul Barry, Nov 03 2010
Equals the INVERTi transform of A055588: (1, 2, 4, 9, 22, 56, ...) - Gary W. Adamson, Apr 01 2011
The Ca3 sums, see A180662, of triangle A194005 equal the terms of this sequence without a(0) and a(1). - Johannes W. Meijer, Aug 16 2011
Without the initial 1, a(n) = row sums of A182097(n)*A007318(n,k); i.e., a Triangular array T(n,k) multiplying the binomial (Pascal's) triangle by the Padovan sequence where a(0) = 1, a(1) = 0 and a(2) = 1. - Bob Selcoe, Jun 28 2013
a(n+1) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 1; 0, 1, 1; 1, 0, 1] or [1, 1, 0; 1, 1, 1; 1, 0, 1] or [1, 1, 1; 1, 1, 0; 0, 1, 1] or [1, 0, 1; 1, 1, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 0, 1; 1, 1, 1; 0, 1, 1] or of the 3 X 3 matrix [1, 1, 0; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
Number of sequences (e(1), ..., e(n-1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) < e(k) and e(i) <= e(k). [Martinez and Savage, 2.8] - Eric M. Schmidt, Jul 17 2017
a(n+1) is the number of words of length n over the alphabet {0,1,2} that do not contain the substrings 01 or 12 and do not start with a 2 and do not end with a 0. - Yiseth K. Rodríguez C., Sep 11 2020

			G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 28*x^6 + 65*x^7 + 151*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Miklos Bona and Rebecca Smith, Pattern avoidance in permutations and their squares, arXiv:1901.00026 [math.CO], 2018. See H(z), Ex. 4.1.
Richard Choulet, Curtz like Transformation
Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From _N. J. A. Sloane_, Feb 01 2013
Stoyan Dimitrov, Sorting by shuffling methods and a queue, arXiv:2103.04332 [math.CO], 2021.
Phan Thuan Do, Thi Thu Huong Tran, and Vincent Vajnovszki, Exhaustive generation for permutations avoiding a (colored) regular sets of patterns, arXiv:1809.00742 [cs.DM], 2018.
Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 18.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 904
H. Magnusson and H. Ulfarsson, Algorithms for discovering and proving theorems about permutation patterns, arXiv preprint arXiv:1211.7110 [math.CO], 2012.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016
Vincent Vatter, Finding regular insertion encodings for permutation classes, arXiv:0911.2683 [math.CO], 2009.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

First differences of A052921.
Cf. A000931, A007318, A055588, A095263, A097550.
Cf. A137531, A159974, A182097, A185963, A194005.

[n le 3 select 1 else 3*Self(n-1)-2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 14 2012

A034943 := proc(n): add(binomial(n+k-1, 3*k), k=0..floor(n/2)) end: seq(A034943(n), n=0..28); # Johannes W. Meijer, Aug 16 2011

LinearRecurrence[{3,-2,1},{1,1,1},30] (* Harvey P. Dale, Aug 11 2017 *)

{a(n) = if( n<1, n = 0-n; polcoeff( (1 - x + x^2) / (1 - 2*x + 3*x^2 - x^3) + x * O(x^n), n), n = n-1; polcoeff( (1 - x + x^2) / (1 - 3*x + 2*x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, Mar 31 2012 */

@CachedFunction
def a(n): # a = A034943
    if (n<3): return 1
    else: return 3*a(n-1) - 2*a(n-2) + a(n-3)
[a(n) for n in range(51)] # G. C. Greubel, Apr 22 2023

a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1, 3*k). - Paul Barry, Jul 06 2004
G.f.: (1 - 2*x)/(1 - 3*x + 2*x^2 - x^3). - Paul Barry, Jul 06 2005
G.f.: 1 + x / (1 - x / (1 - x / (1 - x / (1 + x / (1 - x))))). - Michael Somos, Mar 31 2012
a(-1 - n) = A185963(n). - Michael Somos, Mar 31 2012
a(n) = A095263(n) - 2*A095263(n-1). - G. C. Greubel, Apr 22 2023

Edited by Charles R Greathouse IV, Apr 20 2010

A075427 a(0) = 1; a(n) = a(n-1)+1 if n is even, otherwise a(n) = 2*a(n-1).

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 15, 30, 31, 62, 63, 126, 127, 254, 255, 510, 511, 1022, 1023, 2046, 2047, 4094, 4095, 8190, 8191, 16382, 16383, 32766, 32767, 65534, 65535, 131070, 131071, 262142, 262143, 524286, 524287, 1048574, 1048575, 2097150, 2097151, 4194302, 4194303, 8388606

Offset: 0

Reinhard Zumkeller, Sep 15 2002

Fixed points for permutations A180200, A180201, A180198, and A180199. - Reinhard Zumkeller, Aug 15 2010

The Kn22 sums, see A180662, of triangle A194005 equal the terms of this sequence. - Johannes W. Meijer, Aug 16 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
R. Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A075426, A066880, A083416, A000225 (bisection), A000918 (bisection).

Cf. A180198, A180199, A180200, A180201, A180662, A194005.

a075427 n = a075427_list !! n
a075427_list = 1 : f 1 1 where
   f x y = z : f (x + 1) z where z = (1 + x `mod` 2) * y + 1 - x `mod` 2
-- Reinhard Zumkeller, Feb 27 2012

[2^Floor((n+3)/2)-3/2+(-1)^n/2: n in [0..30]]; // Vincenzo Librandi, Aug 17 2011

A075427 := proc(n) if type(n,'even') then 2^(n/2+1)-1 ; else 2^(1+(n+1)/2)-2 ; end if; end proc: seq(A075427(n), n=0..40); # R. J. Mathar, Feb 18 2011
isA := proc(n) convert(n, base, 2): 1 - %[1] = nops(%) - add(%) end:
select(isA, [$1..4095]); # Peter Luschny, Oct 27 2022

a[0]=1; a[n_]:=a[n]=If[EvenQ[n],a[n-1]+1,2*a[n-1]]; Table[a[n],{n,0,40}] (* Jean-François Alcover, Mar 20 2011 *)
nxt[{n_,a_}]:={n+1,If[OddQ[n],a+1,2a]}; Transpose[NestList[nxt,{0,1},40]][[2]] (* or *) LinearRecurrence[{0,3,0,-2},{1,2,3,6},50] (* Harvey P. Dale, Mar 12 2016 *)

a(n)=2^((n+3)\2)-3/2+(-1)^n/2 \\ Charles R Greathouse IV, Feb 06 2017

def A075427(n): return (1<<(n>>1)+2)-2 if n&1 else (1<<(n>>1)+1)-1 # Chai Wah Wu, Apr 23 2023

a(0) = 1; for n >= 1, a(2*n) = 2^(n+1)-1, a(2*n-1) = 2^(n+1)-2; a(n) = 2^floor((n+3)/2) - 3/2 + (-1)^n/2. - Benoit Cloitre, Sep 17 2002 [corrected by Robert FERREOL, Jan 26 2011]
a(n) = (-1)^n/2 - 3/2 + 2^(n/2)*(1 + sqrt(2) + (1-sqrt(2))*(-1)^n). - Paul Barry, Apr 22 2004
From Paul Barry, Jul 30 2004: (Start)
Interleaved Mersenne numbers: interleaves 2*2^n-1 and 2(2*2^n-1) (A000225(n+1) and 2*A000225(n+1)).
G.f.: (1+2*x)/((1-x^2)*(1-2*x^2));
a(n) = 3*a(n-2) - 2*a(n-4);
a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)). (End)
For n > 0: a(n) = (1 + n mod 2) * a(n-1) + 1 - (n mod 2). - Reinhard Zumkeller, Feb 27 2012
E.g.f.: 2*(cosh(sqrt(2)*x) - sinh(x) + sqrt(2)*sinh(sqrt(2)*x)) - cosh(x). - Stefano Spezia, Jul 11 2023
From Alois P. Heinz, Dec 27 2023: (Start)
a(n) = 2^floor((n+3)/2)-1-(n mod 2).
a(n) = A066880(n) for n>=1. (End)

Formulae corrected and minor edits by Johannes W. Meijer, Aug 16 2011

A103631 Triangle read by rows: T(n,k) = abs(qStirling2(n,k,q)) for q = -1, with 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 2, 1, 0, 1, 1, 4, 3, 3, 1, 0, 1, 1, 5, 4, 6, 3, 1, 0, 1, 1, 6, 5, 10, 6, 4, 1, 0, 1, 1, 7, 6, 15, 10, 10, 4, 1, 0, 1, 1, 8, 7, 21, 15, 20, 10, 5, 1, 0, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1, 0, 1, 1, 10, 9, 36, 28, 56, 35, 35, 15, 6, 1

Offset: 0

Paul Barry, Feb 11 2005

Previous name: An invertible triangle whose row sums are F(n+1).
Triangle inverse has general term (-1)^(n-k)*binomial(floor(n/2),n-k). Diagonal sums are A103632.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 08 2005
Row sums are Fibonacci numbers (A000045).
Another version of triangle in A065941. - Philippe Deléham, Jan 01 2009
From Johannes W. Meijer, Aug 11 2011: (Start)
The T(n,k) coefficients appear in appendix 2 of Parks's remarkable article "A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov" if we assume that the b(r) coefficients are all equal to 1; see the second Maple program.
The T(n,k) triangle is related to a linear (n+1)-th order differential equation with coefficients a(n,k), see triangle A194005.
Parks's triangle appears to be an appropriate name for the triangle given above. (End)

			From _Paul Barry_, Oct 02 2009: (Start)
Triangle begins:
  1,
  0, 1,
  0, 1, 1,
  0, 1, 1, 1,
  0, 1, 1, 2, 1,
  0, 1, 1, 3, 2,  1,
  0, 1, 1, 4, 3,  3,  1,
  0, 1, 1, 5, 4,  6,  3,  1,
  0, 1, 1, 6, 5, 10,  6,  4, 1,
  0, 1, 1, 7, 6, 15, 10, 10, 4, 1
Production matrix is:
  0, 1,
  0, 1, 1,
  0, 0, 0, 1,
  0, 0, 0, 1, 1,
  0, 0, 0, 0, 0, 1,
  0, 0, 0, 0, 0, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (End)

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with corrections.
P. C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov , Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) pp. 694-702.

Cf. A103633 (signed version).

a103631 n k = a103631_tabl !! n !! k
a103631_row n = a103631_tabl !! n
a103631_tabl = [1] : [0,1] : f [1] [0,1] where
   f xs ys = zs : f ys zs where
     zs = zipWith (+)  ([0,0] ++ xs)  (ys ++ [0])
-- Reinhard Zumkeller, May 07 2012

/* As triangle: */ [[Binomial(Floor((2*n-k-1)/2), n-k): k in [0..n]]: n in [0..15]]; // Vincenzo Librandi, Aug 28 2016

From Johannes W. Meijer, Aug 11 2011: (Start)
A103631 := proc(n,k): binomial(floor((2*n-k-1)/2),n-k) end: seq(seq(A103631(n,k), k=0..n), n=0..12);
nmax:=12: for n from 0 to nmax+1 do b(n):=1 od: A103631 := proc(n,k) option remember: local j: if k=0 and n=0 then b(1) elif k=0 and n>=1 then 0 elif k=1 then b(n+1) elif k=2 then b(1)*b(n+1) elif k>=3 then expand(b(n+1)*add(procname(j,k-2), j=k-2..n-2)) fi: end: for n from 0 to nmax do seq(A103631(n,k), k=0..n) od: seq(seq(A103631(n,k),k=0..n), n=0..nmax); # (End)

p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = x; p[x, 2] = x + x^2; p[x_, n_] := p[x, n] = p[x, n - 1] + x^2*p[x, n - 2]; (* with *) Table[ExpandAll[p[x, n]], {n, 0, 10}]; (* or *) a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a] (* Roger L. Bagula, Apr 27 2008 *)
Table[Binomial[Floor[(2*n - k - 1)/2], n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 27 2016 *)
qStirling2[n_, k_, q_] /; 1 <= k <= n := q^(k - 1) qStirling2[n - 1, k - 1, q] + Sum[q^j, {j, 0, k - 1}] qStirling2[n - 1, k, q];
qStirling2[n_, 0, _] := KroneckerDelta[n, 0];
qStirling2[0, k_, _] := KroneckerDelta[0, k];
qStirling2[, , _] = 0;
Table[Abs[qStirling2[n, k, -1]], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 10 2020 *)

from sage.combinat.q_analogues import q_stirling_number2
for n in (0..9):
    print([abs(q_stirling_number2(n,k).substitute(q=-1)) for k in [0..n]])
# Peter Luschny, Mar 09 2020

T(n,k) = binomial(floor((2*n-k-1)/2), n-k).
A polynomial recursion which produces this triangle: p(x, n) = p(x, n - 1) + x^2*p(x, n - 2). - Roger L. Bagula, Apr 27 2008
Sum_{k=0..n} T(n,k)*x^k = A152163(n), A000007(n), A000045(n+1), A026597(n), A122994(n+1), A158608(n), A122995(n+1), A158797(n), A122996(n+1), A158798(n), A158609(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Jun 12 2009
G.f.: (1+(y-1)*x)/(1-x-y^2*x^2). - Philippe Deléham, Mar 09 2012
T(n,k) = T(n-1,k) + T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 09 2012

New name from Peter Luschny, Mar 09 2020

A374439 Triangle read by rows: the coefficients of the Lucas-Fibonacci polynomials. T(n, k) = T(n - 1, k) + T(n - 2, k - 2) with initial values T(n, k) = k + 1 for k < 2.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 3, 4, 1, 1, 2, 4, 6, 3, 2, 1, 2, 5, 8, 6, 6, 1, 1, 2, 6, 10, 10, 12, 4, 2, 1, 2, 7, 12, 15, 20, 10, 8, 1, 1, 2, 8, 14, 21, 30, 20, 20, 5, 2, 1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1, 1, 2, 10, 18, 36, 56, 56, 70, 35, 30, 6, 2

Offset: 0

Peter Luschny, Jul 22 2024

There are several versions of Lucas and Fibonacci polynomials in this database. Our naming follows the convention of calling polynomials after the values of the polynomials at x = 1. This assumes a regular sequence of polynomials, that is, a sequence of polynomials where degree(p(n)) = n. This view makes the coefficients of the polynomials (the terms of a row) a refinement of the values at the unity.
A remarkable property of the polynomials under consideration is that they are dual in this respect. This means they give the Lucas numbers at x = 1 and the Fibonacci numbers at x = -1 (except for the sign). See the example section.
The Pell numbers and the dual Pell numbers are also values of the polynomials, at the points x = -1/2 and x = 1/2 (up to the normalization factor 2^n). This suggests a harmonized terminology: To call 2^n*P(n, -1/2) = 1, 0, 1, 2, 5, ... the Pell numbers (A000129) and 2^n*P(n, 1/2) = 1, 4, 9, 22, ... the dual Pell numbers (A048654).
Based on our naming convention one could call A162515 (without the prepended 0) the Fibonacci polynomials. In the definition above only the initial values would change to: T(n, k) = k + 1 for k < 1. To extend this line of thought we introduce A374438 as the third triangle of this family.
The triangle is closely related to the qStirling2 numbers at q = -1. For the definition of these numbers see A333143. This relates the triangle to A065941 and A103631.

			Triangle starts:
  [ 0] [1]
  [ 1] [1, 2]
  [ 2] [1, 2, 1]
  [ 3] [1, 2, 2,  2]
  [ 4] [1, 2, 3,  4,  1]
  [ 5] [1, 2, 4,  6,  3,  2]
  [ 6] [1, 2, 5,  8,  6,  6,  1]
  [ 7] [1, 2, 6, 10, 10, 12,  4,  2]
  [ 8] [1, 2, 7, 12, 15, 20, 10,  8,  1]
  [ 9] [1, 2, 8, 14, 21, 30, 20, 20,  5,  2]
  [10] [1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1]
.
Table of interpolated sequences:
  |  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |
  |  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|
  |    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |
  |  0 |        -1         |     1   |       1     |       1    |
  |  1 |         1         |     3   |       0     |       4    |
  |  2 |         0         |     4   |       1     |       9    |
  |  3 |         1         |     7   |       2     |      22    |
  |  4 |         1         |    11   |       5     |      53    |
  |  5 |         2         |    18   |      12     |     128    |
  |  6 |         3         |    29   |      29     |     309    |
  |  7 |         5         |    47   |      70     |     746    |
  |  8 |         8         |    76   |     169     |    1801    |
  |  9 |        13         |   123   |     408     |    4348    |

Paolo Xausa, Rows n = 0..150 of the triangle, flattened
Peter Luschny, Illustrating the polynomials.

Triangles related to Lucas polynomials: A034807, A114525, A122075, A061896, A352362.
Triangles related to Fibonacci polynomials: A162515, A053119, A168561, A049310, A374441.
Sums include: A000204 (Lucas numbers, row), A000045 & A212804 (even sums, Fibonacci numbers), A006355 (odd sums), A039834 (alternating sign row).
Type m^n*P(n, 1/m): A000129 & A048654 (Pell, m=2), A108300 & A003688 (m=3), A001077 & A048875 (m=4).
Adding and subtracting the values in a row of the table (plus halving the values obtained in this way): A022087, A055389, A118658, A052542, A163271, A371596, A324969, A212804, A077985, A069306, A215928.
Columns include: A040000 (k=1), A000027 (k=2), A005843 (k=3), A000217 (k=4), A002378 (k=5).
Diagonals include: A000034 (k=n), A029578 (k=n-1), abs(A131259) (k=n-2).
Cf. A029578 (subdiagonal), A124038 (row reversed triangle, signed).
Cf. A039961, A065941 (qStirling2), A103631, A108299, A131259. A133607, A194005, A333143, A374438 (m=3).

function T(n,k) // T = A374439
  if k lt 0 or k gt n then return 0;
  elif k le 1 then return k+1;
  else return T(n-1,k) + T(n-2,k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 23 2025

A374439 := (n, k) -> ifelse(k::odd, 2, 1)*binomial(n - irem(k, 2) - iquo(k, 2), iquo(k, 2)):
# Alternative, using the function qStirling2 from A333143:
T := (n, k) -> 2^irem(k, 2)*qStirling2(n, k, -1):
seq(seq(T(n, k), k = 0..n), n = 0..10);

A374439[n_, k_] := (# + 1)*Binomial[n - (k + #)/2, (k - #)/2] & [Mod[k, 2]];
Table[A374439[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Paolo Xausa, Jul 24 2024 *)

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k > n: return 0
    if k < 2: return k + 1
    return T(n - 1, k) + T(n - 2, k - 2)

from math import comb as binomial
def T(n: int, k: int) -> int:
    o = k & 1
    return binomial(n - o - (k - o) // 2, (k - o) // 2) << o

def P(n, x):
    if n < 0: return P(n, x)
    return sum(T(n, k)*x**k for k in range(n + 1))
def sgn(x: int) -> int: return (x > 0) - (x < 0)
# Table of interpolated sequences
print("|  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |")
print("|  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|")
print("|    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |")
f = "| {0:2d} | {1:9d}         |  {2:4d}   |   {3:5d}     |    {4:4d}    |"
for n in range(10): print(f.format(n, -P(n, -1), P(n, 1), int(2**n*P(n, -1/2)), int(2**n*P(n, 1/2))))

from sage.combinat.q_analogues import q_stirling_number2
def A374439(n,k): return (-1)^((k+1)//2)*2^(k%2)*q_stirling_number2(n+1, k+1, -1)
print(flatten([[A374439(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 23 2025

T(n, k) = 2^k' * binomial(n - k' - (k - k') / 2, (k - k') / 2) where k' = 1 if k is odd and otherwise 0.
T(n, k) = (1 + (k mod 2))*qStirling2(n, k, -1), see A333143.
2^n*P(n, -1/2) = A000129(n - 1), Pell numbers, P(-1) = 1.
2^n*P(n, 1/2) = A048654(n), dual Pell numbers.
T(2*n, n) = (1/2)*(-1)^n*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)*A045721((n-1)/2) ). - G. C. Greubel, Jan 23 2025

A189976 a(n) is the number of incongruent two-color bracelets of n beads, 8 of them black (A005514), having a diameter of symmetry.

Original entry on oeis.org

1, 1, 5, 5, 15, 15, 35, 35, 70, 70, 126, 126, 210, 210, 330, 330, 495, 495, 715, 715, 1001, 1001, 1365, 1365, 1820, 1820, 2380, 2380, 3060, 3060, 3876, 3876, 4845, 4845, 5985, 5985, 7315, 7315, 8855, 8855, 10626

Offset: 8

Vladimir Shevelev, May 03 2011

For n >= 9, a(n-1) is the number of incongruent two-color bracelets of n beads, 9 from them are black (A032281), having a diameter of symmetry.

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
V. Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A005514, A032281, A008805, A058187.

[Binomial(Floor(n/2),4): n in[8..60]]; // Vincenzo Librandi, Aug 10 2014

A189976 :=proc(n): binomial(floor(n/2),4) end: seq(A189976(n), n=8..48); # Johannes W. Meijer, Aug 15 2011

Module[{c=Binomial[Range[4,30],4]},Riffle[c,c]] (* Harvey P. Dale, Aug 09 2014 *)
Table[(Binomial[Floor[n/2], 4]), {n, 8, 40}] (* Vincenzo Librandi, Aug 10 2014 *)

a(n) = C(floor(n/2),4).
a(n+5) = A194005(n,n-4). [Johannes W. Meijer, Aug 15 2011]
G.f.: -x^8/((x-1)^5*(x+1)^4). [Colin Barker, Feb 06 2013]

Data added and link corrected by Johannes W. Meijer, Aug 15 2011

A109222 Row sums of a triangle related to the Fibonacci polynomials.

Original entry on oeis.org

1, 2, 3, 6, 11, 21, 40, 76, 145, 276, 526, 1002, 1909, 3637, 6929, 13201, 25150, 47915, 91286, 173915, 331337, 631252, 1202640, 2291229, 4365172, 8316378, 15844082, 30185609, 57508601, 109563441, 208736561, 397677834, 757642355, 1443434582

Offset: 0

Paul Barry, Jun 22 2005

Row sums of A109221.

The Kn4 sums, see A180662, of triangle A065941 equal the terms of this sequence a(n) while the Kn4 sums of triangle A194005 equal a(n+1)-1. - Johannes W. Meijer, Aug 14 2011

Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

A109222 := proc(n): add(binomial(floor((2*n-k)/2)+n-k, 2*n-2*k), k=0..n) end:  seq(A109222(n), n=0..33); # Johannes W. Meijer, Aug 14 2011

G.f.: (1 + x - x^2 - x^3)/(1 - x - 2x^2 + x^4);
a(n) = a(n-1) + 2a(n-2) - a(n-4);
a(n) = Sum_{k=0..n} binomial(floor((2n-k)/2)+n-k, 2n-2k).

A123029 a(2n-1) = Product_{i=1..n} Fibonacci(2i-1) and a(2n) = Product_{i=1..n} Fibonacci(2i).

Original entry on oeis.org

1, 1, 2, 3, 10, 24, 130, 504, 4420, 27720, 393380, 3991680, 91657540, 1504863360, 55911099400, 1485300136320, 89290025741800, 3838015552250880, 373321597626465800, 25964175210977203200, 4086378207619294646800

Offset: 1

Roger L. Bagula, Sep 25 2006

From Johannes W. Meijer, Aug 21 2011: (Start)
An appropriate name for this sequence is Fibonacci double factorial, cf. A006882.
In Parks' article appendix 2, a number triangle T(n,k) with T(n,n) = a(n+1), n>=0, appears if we assume that b(r) = Fibonacci(r); see A103631 and A194005. (End)
The original name of this sequence was: A000045 inside a second linear differential equation recursion: b(n) = b(n-1) + b(n-2) --> Binet(n) of A000045 a(n) = b(n)*a(n-2)/(n*(n-1)).
Bagula also stated that using the solutions to these second order differential equations Markov/ linear recursions can be encoded as analog functions.
Partial products of the odd-indexed Fibonacci numbers interleaved with the partial products of the even-indexed Fibonacci numbers. - Harvey P. Dale, Mar 14 2012

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 10*x^5 + 24*x^6 + 130*x^7 + 504*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..100
P. C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov, Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) p. 694-702.
Eric Weisstein's World of Mathematics, Fibonorial.

Cf. A000045, A003266, A194157, A194158.

Cf. A006882, A103631, A194005.

function a(n)
  if n lt 2 then return 1;
  else return Fibonacci(n)*a(n-2);
  end if; return a;
end function;
[a(n): n in [1..30]]; // G. C. Greubel, Jul 20 2021

with(combinat): A123029 :=proc(n): if type(n,even) then mul(fibonacci(2*i), i=1..n/2) else mul(fibonacci(2*i-1), i= 1..(n+1)/2) fi: end: seq(A123029(n), n=1..21); # Johannes W. Meijer, Aug 21 2011

a[n_]:= a[n]= If[n<2, 1, Fibonacci[n]*a[n-2]]; Table[a[n], {n, 30}] (* modified by G. C. Greubel, Jul 20 2021 *)
With[{nn=21},Riffle[FoldList[Times,1,Fibonacci[Range[3,nn,2]]],FoldList[ Times,1, Fibonacci[ Range[4,nn+1,2]]]]] (* Harvey P. Dale, Mar 14 2012 *)

{a(n) = if( n<0, 0, prod(k=0, (n-1)\2, fibonacci(n - 2*k)))}; /* Michael Somos, Oct 07 2014 */

def a(n): return 1 if (n<2) else fibonacci(n)*a(n-2)
[a(n) for n in (1..30)] # G. C. Greubel, Jul 20 2021

a(n) = n!*c(n) with c(n) = b(n)*c(n-2)/(n*(n-1)), c(0) = 1, c(1) = 1; b(n) = b(n-1) + b(n-2), b(0) = 0, b(1) = 1 and b(n) = F(n) with F(n) = A000045(n).
From Johannes W. Meijer, Aug 21 2011: (Start)
a(n) = F(n)*a(n-2).
a(2*n) = A194157(n) and a(2*n-1) = A194158(n). (End)
a(0) = 1 by convention since empty products equal 1. - Michael Somos, Oct 07 2014
0 = a(n)*(a(n+2)*a(n+3) - a(n+1)*a(n+4)) + a(n+1)*(+a(n+2)^2) for all n>=0. - Michael Somos, Oct 07 2014

Edited by Johannes W. Meijer, Aug 21 2011

A189980 a(n) is the number of incongruent two-color bracelets of n beads, 10 from them are black (A005515), having a diameter of symmetry.

Original entry on oeis.org

1, 1, 6, 6, 21, 21, 56, 56, 126, 126, 252, 252, 462, 462, 792, 792, 1287, 1287, 2002, 2002, 3003, 3003, 4368, 4368, 6188, 6188, 8568, 8568, 11628, 11628, 15504, 15504, 20349, 20349, 26334, 26334, 33649, 33649

Offset: 10

Vladimir Shevelev, May 03 2011

For n >= 11, a(n-1) is the number of incongruent two-color bracelets of n beads, 11 from them are black (A032282), having a diameter of symmetry.

Hansraj Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
V. Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Cf. A005515, A032282, A008805, A058187, A189976.

A189980 :=proc(n): binomial(floor(n/2),5) end: seq(A189980(n), n=10..47); # Johannes W. Meijer, Aug 15 2011

Table[Binomial[Floor[n/2],5],{n,10,50}] (* Harvey P. Dale, Oct 06 2017 *)

a(n) = binomial(floor(n/2), 5). [Typo fixed by Colin Barker, Feb 07 2013]
a(n+6) = A194005(n, n-5). - Johannes W. Meijer, Aug 15 2011
G.f.: x^10/((x-1)^6*(x+1)^5). - Colin Barker, Feb 07 2013

Data added and link corrected by Johannes W. Meijer, Aug 15 2011

A374441 Triangle read by rows: T(n, k) = binomial(n - floor(k/2), ceiling(k/2)) - binomial(n - ceiling(k/2), floor(k/2)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 4, 0, 3, 0, 0, 0, 5, 0, 6, 0, 1, 0, 0, 6, 0, 10, 0, 4, 0, 0, 0, 7, 0, 15, 0, 10, 0, 1, 0, 0, 8, 0, 21, 0, 20, 0, 5, 0, 0, 0, 9, 0, 28, 0, 35, 0, 15, 0, 1, 0, 0, 10, 0, 36, 0, 56, 0, 35, 0, 6, 0, 0, 0, 11, 0, 45, 0, 84, 0, 70, 0, 21, 0, 1, 0

Offset: 0

Peter Luschny, Jul 19 2024

Member of the family of Fibonacci polynomials (A011973, A162515, ...) and Chebyshev polynomials (A053119).

			Triangle starts:
  [ 0]  0;
  [ 1]  0, 0;
  [ 2]  0, 1, 0;
  [ 3]  0, 2, 0,  0;
  [ 4]  0, 3, 0,  1, 0;
  [ 5]  0, 4, 0,  3, 0,  0;
  [ 6]  0, 5, 0,  6, 0,  1, 0;
  [ 7]  0, 6, 0, 10, 0,  4, 0,  0;
  [ 8]  0, 7, 0, 15, 0, 10, 0,  1, 0;
  [ 9]  0, 8, 0, 21, 0, 20, 0,  5, 0, 0;
  [10]  0, 9, 0, 28, 0, 35, 0, 15, 0, 1, 0;

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with corrections.

Cf. A374440 (odd columns agree).
Cf. A000071 (row sums), A065941, A194005, A103631, A007318.
Cf. A011973, A162515, A053119.

T := (n, k) -> if k::even then 0 else binomial(n - (k + 1)/2, (k + 1)/2) fi:
# Or as a recurrence:
T := proc(n, k) option remember; if k::even or k > n then 0 elif k = 1 then n - 1 else T(n - 1, k) + T(n - 2, k - 2) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..12);

A374441[n_, k_] := If[OddQ[k], Binomial[n - (k + 1)/2, (k + 1)/2], 0];
Table[A374441[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Nov 16 2024 *)

from math import isqrt, comb
def A374441(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    b = n-comb(a+1,2)
    return comb(a-(b+1>>1),b+1>>1) if b&1 else 0 # Chai Wah Wu, Nov 14 2024

from math import comb as binomial
def row(n: int) -> list[int]:
    return [binomial(n - (k+1)//2, (k+1)//2) if k%2 else 0 for k in range(n+1)]
for n in range(11): print(row(n))  # Peter Luschny, Nov 21 2024

T(n, k) = [x^(n-k)][z^n] (x / (1 - x*z - z^2)).
T(n, k) = binomial(n - (k + 1)/2, (k + 1)/2) if k is odd, and otherwise 0.
Sum_{k=0..n} T(n, k) = Fibonacci(n + 1) - 1.
Columns with odd index agree with the odd indexed columns of A374440.

cp's OEIS Frontend

A085478 Triangle read by rows: T(n, k) = binomial(n + k, 2*k).

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A034943 Binomial transform of Padovan sequence A000931.

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A075427 a(0) = 1; a(n) = a(n-1)+1 if n is even, otherwise a(n) = 2*a(n-1).

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A103631 Triangle read by rows: T(n,k) = abs(qStirling2(n,k,q)) for q = -1, with 0 <= k <= n.

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A374439 Triangle read by rows: the coefficients of the Lucas-Fibonacci polynomials. T(n, k) = T(n - 1, k) + T(n - 2, k - 2) with initial values T(n, k) = k + 1 for k < 2.

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A123029 a(2n-1) = Product_{i=1..n} Fibonacci(2i-1) and a(2n) = Product_{i=1..n} Fibonacci(2i).