A026671 -id:A026671 - cp's OEIS frontend

A084868 Main diagonal of symmetric square table A084867, in which the antidiagonal sums (A006012) form the first row shifted left.

1, 2, 8, 36, 168, 796, 3800, 18216, 87536, 421292, 2029592, 9784088, 47187536, 227651352, 1098523504, 5301727824, 25590307552, 123529362124, 596337248024, 2878947861432, 13899229883024, 67105641925064, 323993230750672

Offset: 0

Paul D. Hanna, Jun 10 2003, Jun 11 2003

The Hankel transform (see A001906 for definition) of this sequence is A000302 (powers of 4): 1, 4, 16, 64, 256, 1024, ... - Philippe Deléham, Aug 17 2005

			1 + 2*x + 8*x^2 + 36*x^3 + 168*x^4 + 796*x^5 + 3800*x^6 + 18216*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vincent Pilaud, V Pons, Permutrees, arXiv preprint arXiv:1606.09643, 2016

Cf. A006012, A011782, A028329, A084867, A026671, A081696.

1/(1-x/(sqrt(1/4-x))): series(%,x,23): seq(coeff(%,x,n),n=0..22); # Peter Luschny, Feb 06 2017

Table[SeriesCoefficient[((1-4*x)+2*x*Sqrt[1-4*x])/(1-4*x-4*x^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

{a(n) = if( n<0, 0, polcoeff((1 - 4*x + 2*x * sqrt(1 - 4*x + x * O(x^n))) /(1 - 4*x - 4*x^2), n))} /* Michael Somos, Jan 05 2012 */

Differential equation: (16*x^3 + 12*x^2 - 8*x + 1) * x*(d/dx)A(x) + (8x^3 - 12*x^2 + 6*x - 1) * A(x) + (8x^2 - 6*x + 1) = 0.
G.f.: ((1 - 4*x) + 2*x * sqrt(1 - 4*x)) / (1 - 4*x - 4*x^2). a(n) * (n-1) = a(n-1) * (8*n - 14) - a(n-2) * 12*(n-3) - a(n-3) * 8*(2*n - 5), n > 2. Hankel number wall zig-zag diagonal is A011782. - Michael Somos, Sep 14 2003
INVERT transform of A028329 (offset 1). - Michael Somos, Jan 05 2012
G.f.: (1-2*x*f(x))/(1-2*x*f(x)-2*x) where f(x) is the g.f. of A000108 (Catalan numbers). - Philippe Deléham, Jan 30 2012
a(n) ~ (1-1/sqrt(2))*(2+2*sqrt(2))^n. - Vaclav Kotesovec, Oct 14 2012
From Peter Bala, Feb 05 2017: (Start)
G.f: sqrt(1 - 4*x)/(sqrt(1 - 4*x) - 2*x) =  1/(1 - 2*x/(1 - 2*x/(1 - x/(1 - x/(1 - x/(1 - ...)))))) (continued fraction).  Cf. A026671, A081696.
Catalan transform of A006012, that is, equals A106566*A006012, as noted by R. J. Mathar. (End)

A111959 Renewal array for aerated central binomial coefficients.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 6, 0, 6, 0, 1, 0, 16, 0, 8, 0, 1, 20, 0, 30, 0, 10, 0, 1, 0, 64, 0, 48, 0, 12, 0, 1, 70, 0, 140, 0, 70, 0, 14, 0, 1, 0, 256, 0, 256, 0, 96, 0, 16, 0, 1, 252, 0, 630, 0, 420, 0, 126, 0, 18, 0, 1, 0, 1024, 0, 1280, 0, 640, 0, 160, 0, 20, 0, 1, 924, 0, 2772, 0

Offset: 0

Paul Barry, Aug 23 2005

Row sums are A098615.
Binomial transform (product with C(n,k)) is A111960.
Diagonal sums are A026671 (with interpolated zeros).
Inverse is (1/sqrt(1+4x^2),x/sqrt(1+4x^2)), or (sqrt(-1))^(n-k)*T(n,k). [corrected by Peter Bala, Aug 13 2021]
The Riordan array (1,x/sqrt(1-4*x^2)) is the same array with an additional column of zeros (besides the top element 1) added to the left. - Vladimir Kruchinin, Feb 17 2011

			From _Peter Bala_, Aug 13 2021: (Start)
Triangle begins
  1;
  0,  1;
  2,  0, 1;
  0,  4, 0, 1;
  6,  0, 6, 0, 1;
  0, 16, 0, 8, 0, 1;
Infinitesimal generator begins
  0;
  0, 0;
  2, 0, 0;
  0, 4, 0, 0;
  0, 0, 6, 0, 0;
  0, 0, 0, 8, 0, 0; (End)

Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A054335, A038231, A046521.

Riordan array (1/sqrt(1-4x^2), x/sqrt(1-4x^2)); number triangle T(n, k)=(1+(-1)^(n-k))*binomial((n-1)/2, (n-k)/2)*2^(n-k)/2.
G.f.: 1/(1-xy-2x^2/(1-x^2/(1-x^2/(1-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
From Peter Bala, Aug 13 2021: (Start)
T(2*n,2*k) = A046521(n,k); T(2*n+1,2*k+1) = A038231(n,k).
The row entries, read from right to left, are the coefficients in the n-th order Taylor polynomial of (sqrt(1 + 4*x^2))^((n-1)/2) at x = 0.
The infinitesimal generator of this array has the sequence [2, 4, 6, 8, 10, ...] on the second subdiagonal below the main diagonal and zeros elsewhere.
The m-th power of the array is the Riordan array (1/sqrt(1 - 4*m*x^2), x/sqrt(1 - 4*m*x^2)) with entries given by sqrt(m)^(n-k)*T(n,k). (End)

A111961 Expansion of 1/(sqrt(1-2x-3x^2)-x).

Original entry on oeis.org

1, 2, 6, 18, 56, 176, 558, 1778, 5686, 18230, 58558, 188366, 606588, 1955044, 6305418, 20347342, 65689088, 212146400, 685342218, 2214556478, 7157409064, 23136645472, 74801223162, 241863933094, 782131232390, 2529458676326

Offset: 0

Paul Barry, Aug 23 2005

Row sums of A111960.
A transform of the Fibonacci numbers. - Paul Barry, Sep 23 2005
Apparently the Motzkin transform of (0 followed by A128588). - R. J. Mathar, Dec 11 2008
Inverse binomial transform of A026671. - Philippe Deléham, Feb 11 2009
Hankel transform is 2^n. - Paul Barry, Mar 02 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Moment sequences, transformations, and Spidernet graphs, arXiv:2307.00098 [math.CO], 2023.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 17, 29.
Taras Goy and Mark Shattuck, Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.

Cf. A001006, A026671, A111960, A128588.

CoefficientList[Series[1/(Sqrt[1-2*x-3*x^2]-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C((j-1)/2, (j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2.
a(n) = Sum_{k=0..n} F(k+1)*Sum_{i=0..floor((n-k)/2)} C(n, i)*C(n-i, i+k)/(i+k+1). - Paul Barry, Sep 23 2005
G.f.: M(x)^2/(2*M(x)-M(x)^2), where M(x) is the g.f. of the Motzkin numbers A001006. - Paul Barry, Feb 03 2006
G.f.: 1/(1-2x/(1-x/(1-x^2/(1-x/(1-x/91-x^2/(1-x/(1-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Mar 02 2010
D-finite with recurrence: n*a(n) + (-4*n+3)*a(n-1) + 3*(-n+1)*a(n-2) + 2*(7*n-15)*a(n-3) + 12*(n-3)*a(n-4) = 0. - R. J. Mathar, Nov 15 2012
a(n) ~ (1+sqrt(5))^n / sqrt(5). - Vaclav Kotesovec, Feb 08 2014

A382539 Expansion of 1/(1 - x/(1 - 4*x)^(1/2))^2.

Original entry on oeis.org

1, 2, 7, 28, 117, 498, 2139, 9232, 39953, 173162, 751103, 3259132, 14142973, 61367542, 266223083, 1154592752, 5005724185, 21694354406, 93985418399, 407009142836, 1761880487509, 7623911365210, 32976925264827, 142585750821408, 616281411472257, 2662702949358158

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Cf. A026671, A382540.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 4*x)^(1/2))^2)); // Vincenzo Librandi, May 13 2025

Table[Sum[4^(n-k)* (k+1)* Binomial[n-k/2-1, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, May 13 2025 *)

a(n) = sum(k=0, n, 4^(n-k)*(k+1)*binomial(n-k/2-1, n-k));

a(n) = Sum_{k=0..n} 4^(n-k) * (k+1) * binomial(n-k/2-1,n-k).

D-finite with recurrence (-n+1)*a(n) +2*(4*n-7)*a(n-1) +(-15*n+41)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Apr 02 2025

A032351 Number of permutations of length n which avoid the patterns 2143, 1324 (smooth permutations); or avoid the patterns 1342, 2431; etc.

Original entry on oeis.org

1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248, 44725516, 195354368, 853829272, 3733693872, 16333556838, 71476391800, 312865382004, 1369760107576, 5998008630244, 26268304208032, 115055864102504, 503997820344464, 2207927106851580, 9673223726469136, 42382192892577128, 185702341264971696

Offset: 0

Miklos Bona

			1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 88*x^5 + 366*x^6 + 1552*x^7 + ...

S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.47.
R. P. Stanley, Catalan Numbers, Cambridge, 2015, p. 133.

H. Abe and S. Billey, Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry, 2014.
George Balla, Ghislain Fourier, and Kunda Kambaso, PBW filtration and monomial bases for Demazure modules in types A and C, arXiv:2205.01747 [math.RT], 2022.
C. Bean, M. Tannock and H. Ulfarsson, Pattern avoiding permutations and independent sets in graphs, arXiv:1512.08155 [math.CO], 2015. See Eq. (2).
Christian Bean, Finding structure in permutation sets, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018.
Miklos Bona, The permutation classes equinumerous to the smooth class, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.
M. Bousquet-Mélou and S. Butler, Forest-like permutations, arXiv:math/0603617 [math.CO], 2006.
Rocco Chirivì, Xin Fang, and Ghislain Fourier, Degenerate Schubert varieties in type A, Transformation Groups (2020).
Darla Kremer and Wai Chee Shiu, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013-2014.
Wikipedia, Permutation classes avoiding two patterns of length 4.
A. Woo and A. Yong, When is a Schubert variety Gorenstein?, arXiv:math/0409490 [math.AG], 2004.
A. Woo and A. Yong, When is a Schubert variety Gorenstein?, Advances in Mathematics, Volume 207, Issue 1, 1 December 2006, Pages 205-220.

Cf. A053617.

t1:=(1-5*x+3*x^2+x^2*sqrt(1-4*x))/(1-6*x+8*x^2-4*x^3);
series(t1,x,40);
seriestolist(%); # N. J. A. Sloane, Nov 09 2016

Table[(Sum[(m+3)*(Sum[Sum[2^j*Binomial[j+k, k]*Binomial[m-j, 2*k+1], {j, 0, m-2*k-1}], {k, 0, m/2}]) * Binomial[2*n-m-2, n], {m, 0, n-2}] + Binomial[2*n, n])/(n+1),{n,0,20}] (* Vaclav Kotesovec, Sep 19 2014, after Vladimir Kruchinin *)

a(n):=(sum((m+3)*(sum(sum(2^(j)*binomial(j+k,k)*binomial(m-j,2*k+1),j,0,m-2*k-1),k,0,m/2))*binomial(2*n-m-2,n),m,0,n-2)+binomial(2*n,n))/(n+1); /* Vladimir Kruchinin, Sep 19 2014 */

x='x+O('x^44) /* that many terms */
gf=(1-5*x+3*x^2+x^2*sqrt(1-4*x))/(1-6*x+8*x^2-4*x^3);
Vec(gf) /* show terms */ /* Joerg Arndt, Apr 20 2011 */

G.f.: (1-5*x+3*x^2+x^2*sqrt(1-4*x))/(1-6*x+8*x^2-4*x^3).
G.f.: 1 / (1 - x / (1 - x / (1 - 2*x / (1 - x / (1 - x / (1 - x / (1 - x / ...))))))). - Michael Somos, Apr 18 2012
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) = upper left term in n-th power of the following infinite square production matrix:
  1, 1, 0, 0, 0, 0, ...
  1, 2, 1, 0, 0, 0, ...
  1, 3, 1, 1, 0, 0, ...
  1, 4, 1, 1, 1, 0, ...
  1, 5, 1, 1, 1, 1, ...
  ...
(End)
HANKEL transform is A011782. HANKEL transform of a(n+1) is A011782(n+1). INVERT transform of A026671 with 1 prepended. - Michael Somos, Apr 18 2012
Recurrence: (n-2)*a(n) = 2*(5*n-13)*a(n-1) - 4*(8*n-25)*a(n-2) + 12*(3*n-10)*a(n-3) - 8*(2*n-7)*a(n-4). - Vaclav Kotesovec, Aug 24 2014
a(n) ~ 1/11 * (1 - 5*r + 3*r^2 + r^2*sqrt(1-4*r)) *(25 - 44*r + 24*r^2) / r^n, where r = 1/6*(4 - 2/(-17 + 3*sqrt(33))^(1/3) + (-17 + 3*sqrt(33))^(1/3)) = 0.228155493653961819214572... is the root of the equation -1 + 6*r - 8*r^2 + 4*r^3 = 0. - Vaclav Kotesovec, Aug 24 2014
a(n) = (Sum_{m=0..n-2} (m+3)*(Sum_{k=0..m/2} Sum_{j=0..m-2*k-1} 2^j * binomial(j+k, k) * binomial(m-j, 2*k+1)) * binomial(2*n-m-2,n) + binomial(2*n,n))/(n+1). - Vladimir Kruchinin, Sep 19 2014

More terms from Erich Friedman

A104625 Expansion of 1/(sqrt(1-4*x) - x^2).

Original entry on oeis.org

1, 2, 7, 24, 87, 322, 1211, 4604, 17645, 68042, 263655, 1025632, 4002601, 15662422, 61427543, 241386924, 950160607, 3745589510, 14784496003, 58424093536, 231112008371, 915065382154, 3626113490579, 14379912928572, 57064644495359

Offset: 0

Paul Barry, Mar 17 2005

Diagonal sums of convolution triangle of central binomial coefficients A054335.

Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (2,2). - Alois P. Heinz, Sep 14 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000984, A026671, A054335.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(sqrt(1-4*x) - x^2))); // G. C. Greubel, Aug 12 2018

CoefficientList[Series[1/(Sqrt[1-4*x] -x^2), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)

x='x+O('x^50); Vec(1/(sqrt(1-4*x) - x^2)) \\ G. C. Greubel, Aug 12 2018

Conjecture: n*a(n) + (n-3)*a(n-1) + 2*(-28*n+51)*a(n-2) + 72*(2*n-5)*a(n-3) - n*a(n-4) + (-5*n+3)*a(n-5) + 18*(2*n-5)*a(n-6) = 0. - R. J. Mathar, Feb 20 2015

a(n) = Sum_{k=0..floor(n+2)/2} 4^(n+2-2*k) * binomial(n+1-3*k/2,n+2-2*k). - Seiichi Manyama, Feb 06 2024

A205813 Triangle T(n,k), read by rows, given by (0, 2, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 20, 16, 6, 1, 0, 70, 64, 30, 8, 1, 0, 252, 256, 140, 48, 10, 1, 0, 924, 1024, 630, 256, 70, 12, 1, 0, 3432, 4096, 2772, 1280, 420, 96, 14, 1, 0, 12870, 16384, 12012, 6144, 2310, 640, 126, 16, 1

Offset: 0

Philippe Deléham, Feb 01 2012

Riordan array (1, x/sqrt(1-4*x)). Inverse of Riordan array (1, x*exp(arcsinh(-2*x))).

T is the convolution triangle of the shifted central binomial coefficients binomial(2*(n-1), n-1). - Peter Luschny, Oct 19 2022

			Triangle begins:
  1;
  0,   1;
  0,   2,   1;
  0,   6,   4,   1;
  0,  20,  16,   6,   1;
  0,  70,  64,  30,   8,   1;
  0, 252, 256, 140,  48,  10,   1;

Cf. A054335 and columns listed there.

# Uses function PMatrix from A357368.
PMatrix(10, n -> binomial(2*(n-1), n-1)); # Peter Luschny, Oct 19 2022

T(n,n) = 1 = A000012(n); T(n+1,n) = 2*n = A005843(n); T(n+2,n) = 2*n*(n+2) = A054000(n+1).
Sum_{k=0..n} T(n,k)*x^k = -A081696(n-1), A000007(n), A026671(n-1), A084868(n) for x = -1, 0, 1, 2 respectively.
G.f.: sqrt(1-4*x)/(sqrt(1-4*x)-y*x).
Sum_{k=0..n} T(n,k)*A090192(k) = A000108(n), A000108 = Catalan numbers.

A382515 Expansion of 1/(1 - x/(1 - 4*x)^(5/2)).

Original entry on oeis.org

1, 1, 11, 91, 691, 5101, 37323, 272405, 1987047, 14493479, 105718071, 771148119, 5625136651, 41032826127, 299316769887, 2183389173811, 15926906427179, 116180104751925, 847485191674867, 6182049517420133, 45095462188117951, 328952511222499589, 2399570809473795931

Offset: 0

Seiichi Manyama, Mar 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A026671, A382514.

Cf. A002802.

Table[Sum[4^(n-k)*Binomial[n+3*k/2-1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Mar 30 2025 *)

a(n) = sum(k=0, n, 4^(n-k)*binomial(n+3*k/2-1, n-k));

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n+3*k/2-1,n-k).

D-finite with recurrence (-n+1)*a(n) +2*(12*n-19)*a(n-1) +(-239*n+519)*a(n-2) +2*(638*n-1751)*a(n-3) +1280*(-3*n+10)*a(n-4) +512*(12*n-47)*a(n-5) +2048*(-2*n+9)*a(n-6)=0. - R. J. Mathar, Mar 31 2025

A382540 Expansion of 1/(1 - x/(1 - 4*x)^(1/2))^3.

Original entry on oeis.org

1, 3, 12, 52, 231, 1035, 4650, 20898, 93849, 420935, 1885248, 8430588, 37642819, 167824905, 747143298, 3321632498, 14747814597, 65397373761, 289652172896, 1281454446408, 5663228541975, 25002457308487, 110275917725658, 485935158536874, 2139412626785505

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Cf. A026671, A382539.

Cf. A382542.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 4*x)^(1/2))^3)); // Vincenzo Librandi, May 12 2025

Table[Sum[(4)^(n-k)* Binomial[k+2,2]*Binomial[n-k/2-1, n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *)

a(n) = sum(k=0, n, 4^(n-k)*binomial(k+2, 2)*binomial(n-k/2-1, n-k));

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(k+2,2) * binomial(n-k/2-1,n-k).

D-finite with recurrence 3*(-n+1)*a(n) +6*(6*n-11)*a(n-1) +2*(-71*n+199)*a(n-2) +4*(44*n-183)*a(n-3) +3*(11*n+5)*a(n-4) +2*(-2*n+7)*a(n-5)=0. - R. J. Mathar, Apr 02 2025

A276472 Modified Pascal's triangle read by rows: T(n,k) = T(n-1,k) + T(n-1,k-1), 12. T(n,n) = T(n,n-1) + T(n-1,n-1), n>1. T(1,1) = 1, T(2,1) = 1. n>=1.

Original entry on oeis.org

1, 1, 2, 4, 3, 5, 11, 7, 8, 13, 29, 18, 15, 21, 34, 76, 47, 33, 36, 55, 89, 199, 123, 80, 69, 91, 144, 233, 521, 322, 203, 149, 160, 235, 377, 610, 1364, 843, 525, 352, 309, 395, 612, 987, 1597, 3571, 2207, 1368, 877, 661, 704, 1007, 1599, 2584, 4181

Offset: 1

Yuriy Sibirmovsky, Sep 12 2016

The recurrence relations for the border terms are the only way in which this differs from Pascal's triangle.
Column T(2n,n+1) appears to be divisible by 4 for n>=2; T(2n-1,n) divisible by 3 for n>=2; T(2n,n-2) divisible by 2 for n>=3.
The symmetry of T(n,k) can be observed in a hexagonal arrangement (see the links).
Consider T(n,k) mod 3 = q. Terms with q = 0 show reflection symmetry with respect to the central column T(2n-1,n), while q = 1 and q = 2 are mirror images of each other (see the link).

			Triangle T(n,k) begins:
n\k 1    2    3    4   5    6    7    8    9
1   1
2   1    2
3   4    3    5
4   11   7    8    13
5   29   18   15   21   34
6   76   47   33   36   55   89
7   199  123  80   69   91   144 233
8   521  322  203  149  160  235 377  610
9   1364 843  525  352  309  395 612  987  1597
...
In another format:
__________________1__________________
_______________1_____2_______________
____________4_____3_____5____________
________11_____7_____8_____13________
____29_____18_____15____21_____34____
_76_____47____33_____36____55_____89_

Yuriy Sibirmovsky, T(n,k), read by rows as a linear sequence a(j) for j = 1..5050
Yuriy Sibirmovsky, Symmetrical hexagonal arrangement for initial terms of T(n,k)
Yuriy Sibirmovsky, T(n,k) compared with Pascal's triangle
Yuriy Sibirmovsky, Illustration for T(n,k) mod 3

Cf. A000045, A000204, A001519, A001906, A002878, A005248, A054441, A088305, A122367, A258109, A026671, A026726, A026732, A004187, A049685, A033891, A206351, A092521, A081018, A049684, A058038, A081016, A003482, A049629, A133273, A049664, A049683, A119032, A023039, A007805, A033889.

Nm=12;
T=Table[0,{n,1,Nm},{k,1,n}];
T[[1,1]]=1;
T[[2,1]]=1;
T[[2,2]]=2;
Do[T[[n,1]]=T[[n-1,1]]+T[[n,2]];
T[[n,n]]=T[[n-1,n-1]]+T[[n,n-1]];
If[k!=1&&k!=n,T[[n,k]]=T[[n-1,k]]+T[[n-1,k-1]]],{n,3,Nm},{k,1,n}];
{Row[#,"\t"]}&/@T//Grid

T(n,k) = if (k==1, if (n==1, 1, if (n==2, 1, T(n-1,1) + T(n,2))), if (kMichel Marcus, Sep 14 2016

Conjectures:
Relations with other sequences:
T(n+1,1) = A002878(n-1), n>=1.
T(n,n) = A001519(n) = A122367(n-1), n>=1.
T(n+1,2) = A005248(n-1), n>=1.
T(n+1,n) = A001906(n) = A088305(n), n>=1.
T(2n-1,n) = 3*A054441(n-1), n>=2. [the central column].
Sum_{k=1..n} T(n,k) = 3*A105693(n-1), n>=2. [row sums].
Sum_{k=1..n} T(n,k)-T(n,1)-T(n,n) = 3*A258109(n), n>=2.
T(2n,n+1) - T(2n,n) = A026671(n), n>=1.
T(2n,n-1) - T(2n,n) = 2*A026726(n-1), n>=2.
T(n,ceiling(n/2)) - T(n-1,floor(n/2)) = 2*A026732(n-3), n>=3.
T(2n+1,2n) = 3*A004187(n), n>=1.
T(2n+1,2) = 3*A049685(n-1), n>=1.
T(2n+1,2n) + T(2n+1,2) = 3*A033891(n-1), n>=1.
T(2n+1,3) = 5*A206351(n), n>=1.
T(2n+1,2n)/3 - T(2n+1,3)/5 = 4*A092521(n-1), n>=2.
T(2n,1) = 1 + 5*A081018(n-1), n>=1.
T(2n,2) = 2 + 5*A049684(n-1), n>=1.
T(2n+1,2) = 3 + 5*A058038(n-1), n>=1.
T(2n,3) = 3 + 5*A081016(n-2), n>=2.
T(2n+1,1) = 4 + 5*A003482(n-1), n>=1.
T(3n,1) = 4*A049629(n-1), n>=1.
T(3n,1) = 4 + 8*A119032(n), n>=1.
T(3n+1,3) = 8*A133273(n), n>=1.
T(3n+2,3n+2) = 2 + 32*A049664(n), n>=1.
T(3n,3n-2) = 4 + 32*A049664(n-1), n>=1.
T(3n+2,2) = 2 + 16*A049683(n), n>=1.
T(3n+2,2) = 2*A023039(n), n>=1.
T(2n-1,2n-1) = A033889(n-1), n>=1.
T(3n-1,3n-1) = 2*A007805(n-1), n>=1.
T(5n-1,1) = 11*A097842(n-1), n>=1.
T(4n+5,3) - T(4n+1,3) = 15*A000045(8n+1), n>=1.
T(5n+4,3) - T(5n-1,3) = 11*A000204(10n-2), n>=1.
Relations between left and right sides:
T(n,1) = T(n,n) - T(n-2,n-2), n>=3.
T(n,2) = T(n,n-1) - T(n-2,n-3), n>=4.
T(n,1) + T(n,n) = 3*T(n,n-1), n>=2.

cp's OEIS Frontend

A084868 Main diagonal of symmetric square table A084867, in which the antidiagonal sums (A006012) form the first row shifted left.

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A032351 Number of permutations of length n which avoid the patterns 2143, 1324 (smooth permutations); or avoid the patterns 1342, 2431; etc.

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A205813 Triangle T(n,k), read by rows, given by (0, 2, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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