A011973 -id:A011973 - cp's OEIS frontend

A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 3, 0, 0, 1, 5, 6, 1, 0, 0, 1, 6, 10, 4, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0

Offset: 0

Nadia Heninger and N. J. A. Sloane, May 21 2010

T(n,k) = 0 if k <0 or k > n+1-k.
T(n,k) is the number of binary vectors of length n and weight k containing no pair of adjacent 1's.
Take Pascal's triangle A007318 and push the k-th column downwards by 2k-1 places (k>=1).
Row sums are A000045.
From Emanuele Munarini, May 24 2011: (Start)
Diagonal sums are A000930(n+1).
A sparse subset (or scattered subset) of {1,2,...,n} is a subset never containing two consecutive elements. T(n,k) is the number of sparse subsets of {1,2,...,n} having size k. For instance, for n=4 and k=2 we have the 3 sparse 2-subsets of {1,2,3,4}: 13, 14, 24. (End)
As a triangle, row 2*n-1 consists of the coefficients of Morgan-Voyce polynomial B(n,x), A172431, and row 2*n to the coefficients of Morgan-Voyce polynomial b(n,x), A054142.
Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014
Antidiagonals of the Pascal matrix A007318 read bottom to top, omitting the first antidiagonal. These are also the antidiagonals (omitting the first antidiagonal) read from top to bottom of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Reverse is embedded in A102426. - Tom Copeland, Jul 02 2018

			Triangle begins:
  [1]
  [1, 1]
  [1, 2, 0]
  [1, 3, 1, 0]
  [1, 4, 3, 0, 0]
  [1, 5, 6, 1, 0, 0]
  [1, 6, 10, 4, 0, 0, 0]
  [1, 7, 15, 10, 1, 0, 0, 0]
  [1, 8, 21, 20, 5, 0, 0, 0, 0]
  [1, 9, 28, 35, 15, 1, 0, 0, 0, 0]
  [1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0]
  [1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0]
  [1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0]
  [1, 13, 66, 165, 210, 126, 28, 1, 0, 0, 0, 0, 0, 0]
  [1, 14, 78, 220, 330, 252, 84, 8, 0, 0, 0, 0, 0, 0, 0]
  [1, 15, 91, 286, 495, 462, 210, 36, 1, 0, 0, 0, 0, 0, 0, 0]
  [1, 16, 105, 364, 715, 792, 462, 120, 9, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  ...

Indranil Ghosh, Rows 0..125, flattened
Pantelis A. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Emanuele Munarini and Norma Zagaglia Salvi, Scattered Subsets, Discrete Mathematics 267 (2003), 213-228.
Emanuele. Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Emanuele Munarini, A combinatorial interpretation of the Chebyshev polynomials, SIAM Journal on Discrete Mathematics, Volume 20, Issue 3 (2006), 649-655.
Peter J. Olver, The canonical contact form.
James J. Y. Zhao, Infinite log-concavity and higher order Turán inequality for Speyer's g-polynomial of uniform matroids, arXiv:2409.08085 [math.CO], 2024. See p. 11.

Cf. A000045, A000930, A007318, A011973 (another version), A218272.
All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - N. J. A. Sloane, May 29 2011
A172431 and A054142 describe the odd and even lines of the triangle.

T[n_,k_]:= Binomial[n+1-k,k]; Table[T[n,k],{n,0,12},{k,0,n}]//Flatten (* Stefano Spezia, Sep 16 2024 *)

create_list(binomial(n-k+1,k),n,0,20,k,0,n); /* Emanuele Munarini, May 24 2011 */

T(n,k)=binomial(n+1-k,k) \\ Charles R Greathouse IV, Oct 24 2012

A267633 Expansion of (1 - 4t)/(1 - x + t x^2): a Fibonacci-type sequence of polynomials.

Original entry on oeis.org

1, -4, 1, -4, 1, -5, 4, 1, -6, 8, 1, -7, 13, -4, 1, -8, 19, -12, 1, -9, 26, -25, 4, 1, -10, 34, -44, 16, 1, -11, 43, -70, 41, -4, 1, -12, 53, -104, 85, -20, 1, -13, 64, -147, 155, -61, 4, 1, -14, 76, -200, 259, -146, 24

Offset: 0

Tom Copeland, Jan 18 2016

			Row polynomials:
P(0,t) = 1 - 4t
P(1,t) = 1 - 4t = [-t(0) + (1-4t)] = -t(0) + P(0,t)
P(2,t) = 1 - 5t + 4t^2 = [-t(1-4t) + (1-4t)] = -t P(0,t) + P(1,t)
P(3,t) = 1 - 6t + 8t^2 = [-t(1-4t) + (1-5t+4t^2)] = -t P(1,t) + P(2,t)
P(4,t) = 1 - 7t + 13t^2 - 4t^3 = [-t(1-5t+4t^2) + (1-6t+8t^2)]
P(5,t) = 1 - 8t + 19t^2 - 12t^3 = [-t(1-6t+8t^2) + (1-7t+13t^2)]
P(6,t) = 1 - 9t + 26t^2 - 25t^3 + 4t^4
P(7,t) = 1 - 10t + 34t^2 - 44t^3 + 16t^4
P(8,t) = 1 - 11t + 43t^2 - 70t^3 + 41t^4 - 4t^5
P(9,t) = 1 - 12t + 53t^2 - 104t^3 + 85t^4 - 20t^5
P(10,t) = 1 - 13t + 64t^2 - 147t^3 + 155t^4 - 61t^5 + 4t^6
P(11,t) = 1 - 14t + 76t^2 - 200t^3 + 259t^4 - 146t^5 + 24t^6
...
Apparently: The odd rows for n>1 are reversed rows of A140882 (mod signs), and the even rows for n>0, the 9th, 15th, 21st, 27th, etc. rows of A228785 (mod signs). The diagonals are reverse rows of A202241.

Tom Copeland, Addendum to Elliptic Lie Triad

Cf. A000108, A000297, A008574, A008586, A011973, A022088 A034856, A084103, A098474, A124644, A140882, A202241, A228785.

p = (1 - 4 t) / (1 - x + t x^2) + O[x]^12 // CoefficientList[#, x] &;
CoefficientList[#, t] & /@ p // Flatten (* Andrey Zabolotskiy, Mar 07 2024 *)

O.g.f. G(x,t) = (1 - 4t)/(1 - x + t x^2) = a /  [t (x - (1+sqrt(a))/(2t))(x - (1-sqrt(a))/(2t))] with a = 1-4t.
Recursion P(n,t) = -t P(n-2,t) + P(n-1,t) with P(-1,t)=0 and P(0,t) = 1-4t.
Convolution of the Fibonacci polynomials of signed A011973 Fb(n,-t) with coefficients of (1-4t), e.g., (1-4t)Fb(5,-t) = (1-4t)(1-3t+t^2) = 1-7t+13t^2-4t^3, so, for n>=1 and k<=(n-1), T(n,k) = (-1)^k [-4*binomial(n-(k-1),k-1) - binomial(n-k,k)] with the convention that 1/(-m)! = 0 for m>=1, i.e., let binomial(n,k) = nint[n!/((k+c)!(n-k+c)!)] for c sufficiently small in magnitude.
Third column is A034856, and the fourth, A000297. Embedded in the coefficients of the highest order term of the polynomials is A008586 (cf. also A008574).
With P(0,t)=0, the o.g.f. is H(x,t) = (1-4t) x(1-tx)/[1-x(1-tx)] = (1-4t) Linv(Cinv(tx)/t), where Linv(x) = x/(1-x) with inverse L(x) = x/(1+x) and Cinv(x) = x (1-x) is the inverse of the o.g.f. of the shifted Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2. Then Hinv(x,t) = C[t L(x/(1-4t))]/t = {1 - sqrt[1-4t(x/(1-4t))/[1+x/(1-4t)]]}/2t = {1-sqrt[1-4tx/(1-4t+x)]}/2t = 1/(1-4t) + (-1+t)/(1-4t)^2 x + (1-2t+2t^2)/(1-4t)^3 x^ + (-1+3t-6t^2+5t^3)/(1-4t)^4 + ..., where the numerators are the signed polynomials of A098474, reverse of A124644.
Row sums (t=1) are periodic -3,-3,0,3,3,0, repeat the six terms ... with o.g.f. -3 - 3x (1-x) / [1-x(1-x)]. Cf. A084103.
Unsigned row sums (t=-1) are shifted A022088 with o.g.f. 5 + 5x(1+x) / [x(1+x)].

Data corrected by Andrey Zabolotskiy, Mar 07 2024

A092865 Nonzero elements in Klee's identity Sum[(-1)^k binomial[n,k]binomial[n+k,m],{k,0,n}] == (-1)^n binomial[n,m-n].

Original entry on oeis.org

1, -1, -1, 1, 2, -1, 1, -3, 1, -3, 4, -1, -1, 6, -5, 1, 4, -10, 6, -1, 1, -10, 15, -7, 1, -5, 20, -21, 8, -1, -1, 15, -35, 28, -9, 1, 6, -35, 56, -36, 10, -1, 1, -21, 70, -84, 45, -11, 1, -7, 56, -126, 120, -55, 12, -1, -1, 28, -126, 210, -165, 66, -13, 1, 8, -84, 252, -330, 220, -78, 14, -1, 1, -36, 210, -462, 495

Offset: 0

Eric W. Weisstein, Mar 07 2004

Triangle, with zeros omitted, given by (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 26 2011

Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014

			1;
-1;
-1, 1;
2, -1;
1, -3, 1;
-3, 4, -1;
-1, 6, -5, 1;
4, -10, 6, -1;
Triangle (0, 1, -1, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, ...) begins:
1
0, -1
0, -1, 1
0, 0, 2, -1
0, 0, 1, -3, 1
0, 0, 0, -3, 4, -1
0, 0, 0, -1, 6, -5, 1 ... - _Philippe Deléham_, Dec 26 2011

H.-H. Chern, H.-K. Hwang, T.-H. Tsai, Random unfriendly seating arrangement in a dining table, arXiv preprint arXiv:1406.0614 [math.PR], 2014
T. Copeland, Addendum to Elliptic Lie Triad
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Eric Weisstein's World of Mathematics, Klee's Identity

All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - N. J. A. Sloane, May 29 2011

Flatten[Table[(-1)^n Binomial[n, m-n], {m, 0, 20}, {n, Ceiling[m/2], m}]]

G.f.: 1/(1+y*x+y*x^2). - Philippe Deléham, Feb 08 2012

A152063 Triangle read by rows. Coefficients of the Fibonacci product polynomials F(n) = Product_{k=1..(n - 1)/2} (1 + 4cos^2(kPi/n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 5, 1, 6, 8, 1, 8, 19, 13, 1, 9, 25, 21, 1, 11, 42, 65, 34, 1, 12, 51, 90, 55, 1, 14, 74, 183, 210, 89, 1, 15, 86, 234, 300, 144, 1, 17, 115, 394, 717, 654, 233, 6, 18, 130, 480, 951, 954, 377, 1, 20, 165, 725, 1825, 2622, 1985, 610, 1, 21, 183, 855

Offset: 1

Gary W. Adamson and Roger L. Bagula, Nov 22 2008

The triangle A125076 is formed by reading upward sloping diagonals. - Gary W. Adamson, Nov 26 2008

Bisection of the triangle: odd-indexed rows are reversals of the rows of A126124, even-indexed rows are the reversals of the rows of A123965. - Gary W. Adamson, Aug 15 2010

			First few rows of the triangle are:
1;
1;
1, 2;
1, 3;
1, 5, 5;
1, 6, 8;
1, 8, 19, 13;
1, 9, 25, 21;
1, 11, 42, 65, 34;
1, 12, 51, 90, 55;
1, 14, 74, 183, 210, 89;
1, 15, 86, 234, 300, 144;
1, 17, 115, 394, 717, 654, 233;
1, 18, 130, 480, 951, 954, 377;
1, 20, 165, 725, 1825, 2622, 1985, 610;
1, 21, 183, 855, 2305, 3573, 2939, 987;
...
By row, alternate signs (+,-,+,-,...) with descending exponents. Rows with n terms have exponents (n-1), (n-2), (n-3),...;
Example: There are two rows with 4 terms corresponding to the polynomials
x^3 - 8x^2 + 19x - 13 (roots associated with the heptagon); and
x^3 - 9x^2 + 25x - 21 (roots associated with the 9-gon (nonagon)).

James P. Bradshaw, Philipp Lampe, and Dusan Ziga, Snake graphs and their characteristic polynomials, arXiv:1910.11823 [math.CO], 2019. See 4.7 p. 16.
N. D. Cahill and D. A. Narayan, Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants, Fibonacci Quarterly, 42(3):216-221, 2004.
M. X. He, D. Simon and P. E. Ricci, Dynamics of the zeros of Fibonacci polynomials, Fibonacci Quarterly, 35(2):160-168, 1997.
V. E. Hoggatt and C. T. Long, Divisibility Properties of Generalized Fibonacci Polynomials, Fibonacci Quarterly, 12:113-120, 1974.

Cf. A000045, A002530 (row sums), A125076, A126124, A123965, A011973.

P := proc(n) option remember; if n < 5 then return
ifelse(n < 3, 1, ifelse(n = 3, 1 + 2*q, 1 + 3*q)) fi;
(1 + 3*q)*P(n - 2) - q^2*P(n - 4) end:
T := n -> local k; seq(coeff(P(n), q, k), k = 0..(n-1)/2):
for n from 1 to 12 do T(n) od;  # (after F. Chapoton)  Peter Luschny, May 27 2024
# Alternative:
P := n -> local k; add(binomial(n-k,k)*(1+x)^(floor(n/2)-k)*x^k, k=0..floor(n/2)):
T := n -> local k; seq(coeff(P(n), x, k), k = 0..n/2):
for n from 0 to 12 do T(n) od; # (after F. Chapoton) Peter Luschny, May 28 2024

Recurrence (as monic polynomials) P(n+4) = (1 + 3*q)*P(n+2) - q^2*P(n). - F. Chapoton, May 27 2024

As monic polynomials, these are the numerators of the polynomials from A011973 evaluated at 1/(1+q). - F. Chapoton, May 28 2024

A054123 Right Fibonacci row-sum array T(n,k), n >= 0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 2, 1, 1, 1, 5, 7, 4, 2, 1, 1, 1, 6, 11, 8, 4, 2, 1, 1, 1, 7, 16, 15, 8, 4, 2, 1, 1, 1, 8, 22, 26, 16, 8, 4, 2, 1, 1, 1, 9, 29, 42, 31, 16, 8, 4, 2, 1, 1, 1, 10, 37, 64, 57, 32, 16, 8, 4, 2, 1, 1, 1, 11, 46, 93, 99, 63, 32, 16, 8, 4, 2, 1, 1, 1, 12, 56

Offset: 0

Clark Kimberling

Variant of A052509 with an additional diagonal of 1's. - R. J. Mathar, Oct 12 2011
Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:
  (1)  if x is in g(n-1), then x + 1 is in g(n); and
  (2)  if x is in g(n-1) and x < 2, then x/2 is in g(n).
Then g(1) = {1/1}, g(2) = {1/2,2/1}, g(3) = {1/4,3/2,3/1}, etc. The denominators in g(n) are 2^0, 2^1, ..., 2^(n-1), and T(n,k) is the number of occurrences of 2^(n-1-k), for k = 0..n-1. - Clark Kimberling, Nov 09 2015
G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = (1-x^2*y) / ((1-x*y)*(1-x-x^2*y)). - Jianing Song, May 30 2022

			Rows:
1
1 1
1 1 1
1 2 1 1
1 3 2 1 1
1 4 4 2 1 1
1 5 7 4 2 1 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
S. Sivasubramanian, Signed excedance enumeration in the hyperoctahedral group El. J. Combinat. 21 (1) (2014) #P2.10, Remark 16.
Index entries for triangles and arrays related to Pascal's triangle

Reflection of array in A054124 about vertical central line.
Row sums: 1, 2, 3, 5, 8, 13, ... (Fibonacci numbers, A000045). Central numbers: 1, 1, 2, 4, 8, ... (binary powers, A000079). Cf. A011973.
Cf. A129713.

a054123 n k = a054123_tabl !! n !! k
a054123_row n = a054123_tabl !! n
a054123_tabl = [1] : [1, 1] : f [1] [1, 1] where
   f us vs = ws : f vs ws where
             ws = zipWith (+) (0 : init us ++ [0, 0]) (vs ++ [1])
-- Reinhard Zumkeller, May 26 2015

Clear[t]; t[n_, k_] := t[n, k] = If[k == 0 || k == n || k == n-1, 1, t[n-1, k] + t[n-2, k-1]]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 01 2013 *)

A052509(n,k) = sum(m=0, k, binomial(n-k, m));
T(n,k) = if(k==n, 1, A052509(n-1,k)) \\ Jianing Song, May 30 2022

T(n, 0) = T(n, n) = 1 for n >= 0; T(n, n-1) = 1 for n >= 1; T(n, k) = T(n-1, k) + T(n-2, k-1) for k=1, 2, ..., n-2, n >= 3. [Corrected by Jianing Song, May 30 2022]

T(n, k) = T(n-1, k-1) + U(n-1, k) for k=1, 2, ..., floor(n/2), n >= 3, array U as in A011973.

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 05 2003

A199969 a(n) = the greatest non-divisor h of n (1 < h < n), or 0 if no such h exists.

Original entry on oeis.org

0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Jaroslav Krizek, Nov 26 2011

From Paul Curtz, Feb 09 2015: (Start)
The nonnegative numbers with 0 instead of 1. See A254667(n), which is linked to the Bernoulli numbers A164555(n)/A027642(n), an autosequence of the second kind.
Offset 0 could be chosen.
An autosequence of the second kind is a sequence whose main diagonal is the first upper diagonal multiplied by 2. If the first upper diagonal is
s0, s1, s2, s3, s4, s5, ...,
the sequence is
Ssk(n) = 2*s0, s0, s0 + 2*s1, s0 +3*s1, s0 + 4*s1 + 2*s2, s1 + 5*s1 + 5*s2, etc.
The corresponding coefficients are A034807(n), a companion to A011973(n).
The binomial transform of Ssk(n) is (-1)^n*Ssk(n).
Difference table of a(n):
   0,  0, 2, 3, 4, 5, 6, 7, ...
   0,  2, 1, 1, 1, 1, 1, ...
   2, -1, 0, 0, 0, 0 ...
  -3,  1, 0, 0, 0, ...
   4, -1, 0, 0, ...
  -5,  1, 0, ...
   6, -1, ...
   7, ...
  etc.
a(n) is an autosequence of the second kind. See A054977(n).
The corresponding autosequence of the first kind (a companion) is 0, 0 followed by the nonnegative numbers (A001477(n)). Not in the OEIS.
Ssk(n) = 2*Sfk(n+1) - Sfk(n) where Sfk(n) is the corresponding sequence of the first kind (see A254667(n)).
(End)
Number of binary sequences of length n-1 that contain exactly one 0 and at least one 1. - Enrique Navarrete, May 11 2021

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

Cf. A199968 (the smallest non-divisor h of n (1A199970. A001477, A011973, A034807, A054977, A254667.
Cf. A007978.
Essentially the same as A000027, A028310, A087156 etc.

Join[{0,0},Table[Max[Complement[Range[n],Divisors[n]]],{n,3,70}]] (* or *) Join[{0,0},Range[2,70]] (* Harvey P. Dale, May 31 2014 *)

if(n>2,n-1,0) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = n-1 for n >= 3.

E.g.f.: 1-x^2/2+(x-1)*exp(x). - Enrique Navarrete, May 11 2021

A332602 Tridiagonal matrix M read by antidiagonals: main diagonal is 1,2,2,2,2,..., two adjacent diagonals are 1,1,1,1,1,...

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Mar 06 2020, following a suggestion from Gary W. Adamson

From Gary W. Adamson, Mar 11 2020: (Start)
The upper left entry of M^n gives the Catalan numbers A000108. Extracting 2 X 2, 3 X 3, and 4 X 4 submatrices from M; then generating sequences from the upper left entries of M^n, we obtain the following sequences:
1, 1, 2, 5, 13, ... = A001519 and the convergent is 2.61803... = 2 + 2*cos(2*Pi/5) = (2*cos(Pi/5))^2.
1, 1, 2, 5, 14, 42, 131, ... = A080937 and the convergent is 3.24697... = 2 + 2*cos(2*Pi/7) = (2*cos(Pi/7))^2.
1, 1, 2, 5, 14, 42, 132, 429, 1429, ... = A080938 and the convergent is 3.53208... = 2 + 2*cos(2*Pi/9) = (2*cos(Pi/9))^2. (End)
The characteristic polynomial for the N X N main submatrix M_N is Phi(N, x) = S(N, 2-x) - S(N-1, 2-x), with Chebyshev's S polynomial (see A049310) evaluated at 2-x. Proof by determinant expansion, to obtain the recurrence Phi(N, x) - (x-2)*Phi(N-1, x) - Phi(N-2, x), for N >= 2, and Phi(0, x) = 1 and Phi(1, x) = 1 - x, that is Phi(-1, x) = 1. The trace is tr(M_N) = 1 + 2^(N-1) = A000051(N-1), and Det(M_N) = 1. - Wolfdieter Lang, Mar 13 2020
The explicit form of the characteristic polynomial for the N X N main submatrix M_N is Phi(N, x) := Det(M_N - x*1_N) = Sum_{k=0..N} binomial(N+k, 2*k)*(-x)^k = Sum_{k=0..N} A085478(N, k)*(-x)^k, for N >= 0, with Phi(0, x) := 1. Proof from the recurrence given in the preceding comment. - Wolfdieter Lang, Mar 25 2020
For the proofs of the 2 X 2, 3 X 3 and 4 X 4 conjectures, see the comments in the respective A-numbers A001519, A080937 and A080938. - Wolfdieter Lang, Mar 30 2020
Replace the main diagonal 1,2,2,2,... of the matrix M with 1,0,0,0,...; 1,1,1,1,...; 1,3,3,3,...; 1,2,1,2,...; 1,2,3,4,...; 1,0,1,0...; and 1,1,0,0,1,1,0,0,.... Take powers of M and extract the upper left terms, resulting in respectively: A001405, A001006, A033321, A176677, A006789, A090344, and A007902. - Gary W. Adamson, Apr 12 2022
The statement that the upper left entry of M^n is a Catalan number is equivalent to Exercise 41 of R. Stanley, "Catalan Numbers." - Richard Stanley, Feb 28 2023
If the upper left 1 in matrix M is replaced with 3, taking powers of the resulting matrix and extracting the upper left terms apparently results in sequence A001700. - Gary W. Adamson, Apr 03 2023

			The matrix begins:
  1, 1, 0, 0, 0, ...
  1, 2, 1, 0, 0, ...
  0, 1, 2, 1, 0, ...
  0, 0, 1, 2, 1, ...
  0, 0, 0, 1, 2, ...
  ...
The first few antidiagonals are:
  1;
  1, 1;
  0, 2, 0;
  0, 1, 1, 0;
  0, 0, 2, 0, 0;
  0, 0, 1, 1, 0, 0;
  0, 0, 0, 2, 0, 0, 0;
  0, 0, 0, 1, 1, 0, 0, 0;
  0, 0, 0, 0, 2, 0, 0, 0, 0;
  0, 0, 0, 0, 1, 1, 0, 0, 0, 0;
  ...
Characteristic polynomial of the 3 X 3 matrix M_3: Phi(3, x) = 1 - 6*x + 5*x^2 - x^3, from {A085478(3, k)}_{k=0..3} = {1, 6, 5, 1}. - _Wolfdieter Lang_, Mar 25 2020

Richard P. Stanley, "Catalan Numbers", Cambridge University Press, 2015.

Cf. A000108, A000051, A001519, A049310, A080937, A080938, A085478.
Cf. A001006, A001405, A006789, A033321, A176677, A090344, A007902.
Cf. A001333 (permanent of the matrix M).
Cf. A054142, A053123, A011973 (characteristic polynomials of submatrices of M).
Cf. A001700.

A060083 Coefficients of even-indexed Euler polynomials (rising powers without zeros).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -3, 5, -3, 1, 17, -28, 14, -4, 1, -155, 255, -126, 30, -5, 1, 2073, -3410, 1683, -396, 55, -6, 1, -38227, 62881, -31031, 7293, -1001, 91, -7, 1, 929569, -1529080, 754572, -177320, 24310, -2184, 140, -8, 1, -28820619

Offset: 0

Wolfdieter Lang, Mar 29 2001

E(2*n,1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. Cigler, q-Fibonacci polynomials and q-Genocchi numbers, arXiv:0908.1219
H.-C. Herbig, D. Herden, C. Seaton, On compositions with x^2/(1-x), arXiv preprint arXiv:1404.1022, 2014

A060082 (falling powers).
Matrix inverse is A102054. Column 0 is A001469 (Genocchi numbers).
Cf. A102054, A001469.

t[n_, k_] := Binomial[2*n, 2*k]*2*(n - k)*EulerE[2*(n - k) - 1, 0]/(2*k + 1); t[n_, n_] = 1; Table[t[n, k], {n, 0, 9}, {k, 0, n }] // Flatten (* Jean-François Alcover, Jul 03 2013 *)

{T(n,k)=local(X=x+x*O(x^(2*n)),Y=y+y*O(y^(2*k+1))); (2*n)!*polcoeff(polcoeff((cosh(X*Y)*(Y-1)+ exp(X*Y)/(exp(X)+1)+exp(-X*Y)/(exp(-X)+1))/Y,2*n,x),2*k,y)} (Hanna)

E(2*n, x)= sum(a(n, m)*x^(2*m+1), m=0..n-1) + x^(2*n), n >= 1; E(0, x)=1.
T(n, k) = A102054(n, k+1) - A102054(n+1, k+1), where A102054 is matrix inverse. E.g.f.: A(x^2, y^2) = [cosh(xy)*(y-1) + exp(xy)/(exp(x)+1) + exp(-xy)/(exp(-x)+1)]/y. - Paul D. Hanna, Dec 28 2004
T(n,k) = 1/(2*k+1)*binomial(2*n,2*k)*A001469(n-k) for 0 <= k <= n-1.
Let F(n,x) = Sum_{k=0..n-1} binomial(n-k-1,k)*x^k be a Fibonacci polynomial (see A011973 for coefficients). Then F(2*n,x) = -Sum_{k=0..n-1} T(n,k)*F(2*k+1,x). For example, F(8,x) = -17*F(1,x) + 28*F(3,x) - 14*F(5,x) + 4*F(7,x). See Cigler, Corollary 1.3. - Peter Bala, Mar 14 2012

A112576 A Chebyshev-related transform of the Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 4, 6, 16, 29, 67, 132, 288, 588, 1253, 2597, 5480, 11430, 24020, 50233, 105383, 220632, 462528, 968808, 2030377, 4253641, 8913436, 18675174, 39131464, 81989909, 171795691, 359958780, 754224480, 1580315220, 3311234189

Offset: 0

Paul Barry, Sep 14 2005

Transform of the Fibonacci numbers by the Chebyshev related transform which maps g(x) -> (1/(1-x^2))g(x/(1-x^2)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Huylebrouck, The Meta-Golden Ratio Chi, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture.
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).

Cf. A011973, A192232.

a:=[0,1,1,4];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2]-a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jul 29 2019

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 29 2019

(* see A192232 for Mmca code. - M. F. Hasler, Apr 05 2016 *)
LinearRecurrence[{1,3,-1,-1},{0,1,1,4},40] (* Harvey P. Dale, May 09 2025 *)

Vec(x/(1-x-3*x^2+x^3+x^4)+O(x^40)) \\ M. F. Hasler, Apr 05 2016

(x/(1-x-3*x^2+x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019

G.f.: x/(1-x-3*x^2+x^3+x^4).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*F(n-2*k).
a(n) = Sum_{k=0..n} C((n+k)/2, k)*(1+(-1)^(n-k))*F(k)/2.
a(n) = (Fibonacci(n+1, (1+sqrt(5))/2) - Fibonacci(n+1, (1-sqrt(5))/2) )/sqrt(5), where Fibonacci(n,x) is the Fibonacci polynomial (see A011973). - G. C. Greubel, Jul 29 2019

A180182 Triangle read by rows: T(n,k) is the number of compositions of n without 7's and having k parts; 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 1, 5, 21, 35, 35, 21, 7, 1, 1, 6, 25, 56, 70, 56, 28, 8, 1, 1, 7, 30, 80, 126, 126, 84, 36, 9, 1, 1, 8, 36, 108, 205, 252, 210, 120, 45, 10, 1, 1, 9, 43, 141, 310, 456, 462, 330, 165, 55, 11, 1

Offset: 1

Emeric Deutsch, Aug 15 2010

			T(10,2)=7 because we have (1,9), (9,1), (2,8), (8,2), (6,4), (4,6), and (5,5).
Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  6,  4,  1;
  1,  5, 10, 10,  5,  1;
  0,  6, 15, 20, 15,  6,  1;

P. Chinn and S. Heubach, Compositions of n with no occurrence of k, Congressus Numerantium, 164 (2003), pp. 33-51 (see Table 9).
R.P. Grimaldi, Compositions without the summand 1, Congressus Numerantium, 152, 2001, 33-43.

P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.

Cf. A011973, A180177, A180178, A180179, A180180, A180181, A180183.

p := 7: T := proc (n, k) options operator, arrow: sum((-1)^(k-j)*binomial(k, j)*binomial(n-p*k+p*j-1, j-1), j = (p*k-n)/(p-1) .. k) end proc: for n to 13 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form
p := 7: g := z/(1-z)-z^p: G := t*g/(1-t*g): Gser := simplify(series(G, z = 0, 15)): for n to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 13 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form
with(combinat): m := 7: T := proc (n, k) local ct, i: ct := 0: for i to numbcomp(n, k) do if member(m, composition(n, k)[i]) = false then ct := ct+1 else end if end do: ct end proc: for n to 12 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

p = 7; max = 14; g = z/(1-z) - z^p; G = t*g/(1-t*g); Gser = Series[G, {z, 0, max+1}]; t[n_, k_] := SeriesCoefficient[Gser, {z, 0, n}, {t, 0, k}]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 28 2014, after Maple *)

Number of compositions of n without p's and having k parts = Sum_{j=(pk-n)/(p-1)..k} (-1)^(k-j)*binomial(k,j)*binomial(n-pk+pj-1, j-1).

For a given p, the g.f. of the number of compositions without p's is G(t,z) = t*g(z)/(1-t*g(z)), where g(z) = z/(1-z) - z^p; here z marks sum of parts and t marks number of parts.

cp's OEIS Frontend

A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

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A267633 Expansion of (1 - 4t)/(1 - x + t x^2): a Fibonacci-type sequence of polynomials.

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A092865 Nonzero elements in Klee's identity Sum[(-1)^k binomial[n,k]binomial[n+k,m],{k,0,n}] == (-1)^n binomial[n,m-n].

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A152063 Triangle read by rows. Coefficients of the Fibonacci product polynomials F(n) = Product_{k=1..(n - 1)/2} (1 + 4*cos^2(k*Pi/n)).

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A054123 Right Fibonacci row-sum array T(n,k), n >= 0, 0<=k<=n.

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A199969 a(n) = the greatest non-divisor h of n (1 < h < n), or 0 if no such h exists.

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A332602 Tridiagonal matrix M read by antidiagonals: main diagonal is 1,2,2,2,2,..., two adjacent diagonals are 1,1,1,1,1,...

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A060083 Coefficients of even-indexed Euler polynomials (rising powers without zeros).

A152063 Triangle read by rows. Coefficients of the Fibonacci product polynomials F(n) = Product_{k=1..(n - 1)/2} (1 + 4cos^2(kPi/n)).