A033321 -id:A033321 - cp's OEIS frontend

A363810 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-14-3, and 4-5-3-1-2.

1, 1, 2, 6, 21, 79, 306, 1196, 4681, 18308, 71564, 279820, 1095533, 4298463, 16913428, 66769536, 264526329, 1051845461, 4197832133, 16813161765, 67571221016, 272448598737, 1101876945673, 4469106749281, 18174503562880, 74093063050412, 302753929958872

Offset: 0

Andrei Asinowski, Jun 23 2023

nonn

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-14-3 and 2-1-3-5-4.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "5" and "8". See the Merino and Mütze reference, Table 3, entry "123458".

Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363811, A363812, A363813, A006012.

with(gfun): seq(coeff(algeqtoseries(x^8*(-2+x)^2*F^4 - x^3*(x-1)*(-2+x)*(x^5-7*x^4+4*x^3-6*x^2+5*x-1)*F^3 - x*(x-1)*(4*x^7-22*x^6+37*x^5-42*x^4+53*x^3-35*x^2+10*x-1)*F^2 - (5*x^6-16*x^5+15*x^4-28*x^3+23*x^2-8*x+1)*(x-1)^2*F - (2*x^5-5*x^4+4*x^3-10*x^2+6*x-1)*(x-1)^2, x, F, 32, true)[1], x, n+1), n = 0..30); # Vaclav Kotesovec, Jun 24 2023

The generating function F=F(x) satisfies the equation x^8*(x - 2)^2*F^4 - x^3*(x - 1)*(x - 2)*(x^5 - 7*x^4 + 4*x^3 - 6*x^2 + 5*x - 1)*F^3 - x*(x - 1)*(4*x^7 - 22*x^6 + 37*x^5 - 42*x^4 + 53*x^3 - 35*x^2 + 10*x - 1)*F^2 - (5*x^6 - 16*x^5 + 15*x^4 - 28*x^3 + 23*x^2 - 8*x + 1)*(x - 1)^2*F - (2*x^5 - 5*x^4 + 4*x^3 - 10*x^2 + 6*x - 1)*(x - 1)^2 = 0.

A363811 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-3-5-4, and 4-5-3-1-2.

Original entry on oeis.org

1, 1, 2, 6, 22, 88, 362, 1488, 6034, 24024, 93830, 359824, 1357088, 5043260, 18501562, 67120024, 241169322, 859450004, 3041415520, 10699090888, 37448249502, 130518538696, 453276141238, 1569476495000, 5420784841936, 18683861676756, 64286814548706

Offset: 0

Andrei Asinowski, Jun 23 2023

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-3-5-4 and 4-5-3-1-2.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "7" and "8". See the Merino and Mütze reference, Table 3, entry "123478".

Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (18,-141,630,-1767,3224,-3834,2896,-1312,320,-32).

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363812, A363813, A006012.

CoefficientList[Series[(1 - x)*(1 - 16*x + 109*x^2 - 410*x^3 + 923*x^4 - 1256*x^5 + 988*x^6 - 400*x^7 + 66*x^8 - 2*x^9)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2*(1 - 2*x)^4),{x,0,26}],x] (* Stefano Spezia, Jun 24 2023 *)

G.f.: (1 - x)*(1 - 16*x + 109*x^2 - 410*x^3 + 923*x^4 - 1256*x^5 + 988*x^6 - 400*x^7 + 66*x^8 - 2*x^9)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2*(1 - 2*x)^4).

A363812 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-4-3, and 3-41-2.

Original entry on oeis.org

1, 1, 2, 6, 20, 69, 243, 870, 3159, 11611, 43130, 161691, 611065, 2325739, 8907360, 34304298, 132770564, 516164832, 2014739748, 7892775473, 31022627947, 122304167437, 483513636064, 1916394053725, 7613498804405, 30313164090695

Offset: 0

Andrei Asinowski, Jun 23 2023

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-4-3 and 3-41-2.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "5", "6", "7". See the Merino and Mütze reference, Table 3, entry "1234567".

Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363811, A363813, A006012.

CoefficientList[Series[(1 - 3*x + 3*x^2 - Sqrt[1 - 6*x + 7*x^2 + 2*x^3 + x^4])/(2*x^2*(2 - x)),{x,0,25}],x] (* Stefano Spezia, Jun 24 2023 *)

G.f.: (1 - 3*x + 3*x^2 - sqrt(1 - 6*x + 7*x^2 + 2*x^3 + x^4))/(2*x^2*(2 - x)).

A363813 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-4-3, and 4-5-3-1-2.

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 295, 1114, 4166, 15390, 56167, 202738, 724813, 2570276, 9052494, 31702340, 110503497, 383691578, 1328039043, 4584708230, 15793983638, 54315199642, 186526735307, 639831906594, 2192754259993, 7509139583560, 25699765092254, 87913948206096

Offset: 0

Andrei Asinowski, Jun 23 2023

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-4-3 and 4-5-3-1-2.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "5", "7", "8". See the Merino and Mütze reference, Table 3, entry "1234578".

Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (10,-37,62,-47,16,-2).

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363811, A363812, A006012.

CoefficientList[Series[(1 - 9*x + 29*x^2 - 39*x^3 + 20*x^4 - 3*x^5)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2),{x,0,27}],x] (* Stefano Spezia, Jun 24 2023 *)

G.f.: (1 - 9*x + 29*x^2 - 39*x^3 + 20*x^4 - 3*x^5)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2).

A129167 Number of base pyramids in all skew Dyck paths of semilength n.

Original entry on oeis.org

0, 1, 3, 9, 30, 109, 420, 1685, 6960, 29391, 126291, 550359, 2426502, 10803801, 48507843, 219377949, 998436792, 4569488371, 21016589073, 97090411019, 450314942682, 2096122733211, 9788916220518, 45850711498859, 215348942668680, 1013979873542689, 4785437476592805, 22633143884165985, 107258646298581390

Offset: 0

Emeric Deutsch, Apr 04 2007

nonn

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A base pyramid is a pyramid starting on the x-axis.
a(n) = |A091699(n+1)|. Partial sums of A033321(n), n = 1, 2, 3, ....
a(n+1) is the number of 3-colored Motzkin paths of length n with no peaks at level 1. - José Luis Ramírez Ramírez, Mar 31 2013

			a(2)=3 because in the paths (UD)(UD), (UUDD) and UUDL we have altogether 3 base pyramids (shown between parentheses).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Cf. A033321, A091699, A129165.

G:=(1-3*z-sqrt(1-6*z+5*z^2))/z/(3-3*z-sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);

CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2])/(x*(3-3*x-Sqrt[1-6*x+5*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

z='z+O('z^66); concat([0], Vec((1-3*z-sqrt(1-6*z+5*z^2))/z/(3-3*z-sqrt(1-6*z+5*z^2)))) \\ Joerg Arndt, Aug 27 2014

a(n) = Sum_{k=0..n} k*A129165(n,k).
G.f.: (1 - 3*z - sqrt(1 - 6*z + 5*z^2))/(z*(3 - 3*z - sqrt(1 - 6*z + 5*z^2))).
Recurrence: 2*(n+1)*a(n) = (13*n-3)*a(n-1) - 4*(4*n-3)*a(n-2) + 5*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 5^(n+5/2)/(72*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012

A171224 Riordan array (f(x),x*f(x)) where f(x) is the g.f. of A117641.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 11, 6, 3, 0, 1, 42, 23, 9, 4, 0, 1, 167, 90, 36, 12, 5, 0, 1, 684, 365, 144, 50, 15, 6, 0, 1, 2867, 1518, 595, 204, 65, 18, 7, 0, 1, 12240, 6441, 2511, 858, 270, 81, 21, 8, 0, 1, 53043, 27774, 10782, 3672, 1155, 342, 98, 24, 9, 0, 1

Offset: 0

Philippe Deléham, Dec 05 2009

			Triangle begins
    1;
    0,  1;
    1,  0,  1;
    3,  2,  0,  1;
   11,  6,  3,  0,  1;
   42, 23,  9,  4,  0,  1;
  167, 90, 36, 12,  5,  0,  1;
  ...
Production array begins
    0,  1;
    1,  0,  1;
    3,  1,  0,  1;
    9,  3,  1,  0,  1;
   27,  9,  3,  1,  0,  1;
   81, 27,  9,  3,  1,  0,  1;
  243, 81, 27,  9,  3,  1,  0,  1;
  ... - _Philippe Deléham_, Mar 04 2013

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

Cf. A053121, A097609, A065600.

[[((k+1)/(n+1))*(&+[3^(n-k-2*j)*Binomial(n+1,j)*Binomial(n-k-j-1, n-k-2*j): j in [0..Floor((n-k)/2)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 04 2019

T[n_, k_]:= (k+1)/(n+1)*Sum[3^(n-k-2*j)*Binomial[n+1,j]*Binomial[n-k-j-1, n-k-2*j], {j, 0, Floor[(n-k)/2]}]; Table[T[n, k], {n,0,10}, {k,0,n} ]//Flatten (* G. C. Greubel, Apr 04 2019 *)

T(n,k):=(k+1)/(n+1)*sum(3^(n-k-2*j)*binomial(n+1,j)*binomial(n-k-j-1,n-k-2*j),j,0,floor((n-k)/2)); /* Vladimir Kruchinin, Apr 04 2019 */

{T(n,k) = ((k+1)/(n+1))*sum(j=0, floor((n-k)/2), 3^(n-k-2*j) *binomial(n+1,j)*binomial(n-k-j-1, n-k-2*j))}; \\ G. C. Greubel, Apr 04 2019

[[((k+1)/(n+1))*sum(3^(n-k-2*j)*binomial(n+1,j)*binomial(n-k-j-1, n-k-2*j) for j in (0..floor((n-k)/2))) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 04 2019

Sum_{k=0..n} T(n,k)*x^k = A117641(n), A033321(n), A007317(n+1), A002212(n+1), A026378(n+1) for x = 0, 1, 2, 3, 4 respectively.
Triangle equals B*A065600*B^(-1) = B^2*A097609*B^(-2) = B^3*A053121*B^(-3), product considered as infinite lower triangular arrays and B = A007318. - Philippe Deléham, Dec 08 2009
T(n,k) = T(n-1,k-1) + Sum_{i>=0} T(n-1,k+1+i)*3^i, T(0,0) = 1. - Philippe Deléham, Feb 23 2012
T(n,k) = ((k+1)/(n+1))*Sum_{j=0..floor((n-k)/2)} 3^(n-k-2*j)*C(n+1,j)*C(n-k-j-1,n-k-2*j). - Vladimir Kruchinin, Apr 04 2019

Terms a(55) onward added by G. C. Greubel, Apr 04 2019

A154930 Inverse of Fibonacci convolution array A154929.

Original entry on oeis.org

1, -2, 1, 5, -4, 1, -15, 14, -6, 1, 51, -50, 27, -8, 1, -188, 187, -113, 44, -10, 1, 731, -730, 468, -212, 65, -12, 1, -2950, 2949, -1956, 970, -355, 90, -14, 1, 12235, -12234, 8291, -4356, 1785, -550, 119, -16, 1, -51822, 51821, -35643, 19474, -8612, 3021

Offset: 0

Paul Barry, Jan 17 2009

Alternating sign version of A104259. Row sums are (-1)^n*A033321. First column is (-1)^n*A007317.

			Triangle begins
1,
-2, 1,
5, -4, 1,
-15, 14, -6, 1,
51, -50, 27, -8, 1,
-188, 187, -113, 44, -10, 1,
731, -730, 468, -212, 65, -12, 1,
-2950, 2949, -1956, 970, -355, 90, -14, 1
Production array is
-2, 1,
1, -2, 1,
-1, 1, -2, 1,
1, -1, 1, -2, 1,
-1, 1, -1, 1, -2, 1,
1, -1, 1, -1, 1, -2, 1,
-1, 1, -1, 1, -1, 1, -2, 1
or ((1-x-x^2)/(1+x),x) beheaded.

Cf. A171567, A054336, A171488, A035324.

Riordan array ((1/(1+x))c(-x/(1+x)), (x/(1+x))c(x/(1+x))), c(x) the g.f. of A000108;
Riordan array ((sqrt(1+6x+5x^2)-x-1)/(2x(1+x)),(sqrt(1+6x+5x^2)-x-1)/ (2(1+x)));
Triangle T(n,k) = sum{j=0..n, (-1)^(n-k)*C(n,j)*C(2j-k,j-k)(k+1)/(j+1)}.
T(n,k) = T(n-1,k-1) -2*T(n-1,k) + Sum_{i, i>=0} T(n-1,k+1+i)*(-1)^i. - Philippe Deléham, Feb 23 2012

A171380 Expansion of the first column of triangle T_(1,x), T(x,y) defined in A039599; T_(1,0)= A061554, T_(1,1)= A064189, T_(1,2)= A039599, T_(1,3)= A110877, T_(1,4)= A124576.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 6, 2, 1, 0, 0, 10, 8, 2, 1, 0, 0, 20, 16, 12, 2, 1, 0, 0, 35, 47, 25, 17, 2, 1, 0, 0, 70, 94, 97, 36, 23, 2, 1, 0, 0, 126, 244, 204, 179, 49, 30, 2, 1, 0, 0

Offset: 0

Philippe Deléham, Dec 07 2009

Diagonal sums: A089324.

Equal to A092107*B^(-1) = A092107*A130595 as lower triangular arrays. - Philippe Deléham, Dec 10 2009

			Triangle begins:
   1;
   1, 0;
   2, 0, 0;
   3, 1, 0, 0;
   6, 2, 1, 0, 0;
  10, 8, 2, 1, 0, 0;
  ...

Cf. A000108, A001006, A001405, A033321, A089324.

Sum_{k=0..n} T(n,k)*x^k = A001405(n), A001006(n), A000108(n), A033321(n) for x = 0, 1, 2, 3 respectively.

A278301 Number of permutations of length n in the class of juxtapositions of 321-avoiders with 21-avoiders.

Original entry on oeis.org

1, 1, 2, 6, 23, 98, 434, 1949, 8803, 39888, 181201, 825201, 3767757, 17249560, 79191480, 364585230, 1683208515, 7792546040, 36174065285, 168367375735, 785637327745, 3674914227120, 17230132657815, 80965662243526, 381275131584373, 1799105397745998

Offset: 0

Jakub Sliacan, Nov 17 2016

nonn

a(n) is also the number of permutations of length n in the class of juxtapositions of 231-avoiders with 21-avoiders.

			There are 23 permutations of length 4 which can be expressed as a juxtaposition of a 321-avoider and a 21-avoider. Only 4321 is not expressable this way.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Robert Brignall, Jakub Sliacan, Juxtaposing Catalan permutation classes with monotone ones, arXiv:1611.05370 [math.CO], 2016.
Robert Brignall, Jakub Sliacan, Combinatorial specifications for juxtapositions of permutation classes, arXiv:1902.02705 [math.CO], 2019.

The other two juxtapositions of Catalan and monotone classes are enumerated by A033321, A165538.

e = ee /. Solve[ee == 1 + x/(1 - x) ee, ee][[1]];
c = cc /. Solve[cc == 1 + x cc^2, cc][[1]];
cb = ccb /. Solve[ccb == 1 + x/(1 - x) ccb^2, ccb][[2]];
b = bb /. Solve[bb == x^2/(1 - x) + x c bb e, bb][[1]];
m = mm /.
   Solve[mm ==
      x c mm cb + b e x/(1 - x) (cb - 1) + x^2/(1 - x) (cb - 1),
     mm][[1]];
f = c + c m cb/(1 - x);
CoefficientList[Series[f, {x, 0, 25}], x]
Rest[CoefficientList[Series[(1 - (1 - 4 x)^(1/2) + x (-4 + (1 - 4 x)^(1/2) + ((-1 + 5 x)^(1/2)) / ((-1 + x)^(1/2))))/ (-2 x^2), {x, 0, 33}], x]] (* Vincenzo Librandi, Nov 18 2016 *)

G.f.: (1 - (1-4*x)^(1/2) + x*(-4 + (1-4*x)^(1/2) + ((-1+5*x)^(1/2)) / ((-1+x)^(1/2)))) / (-2*x^2).

a(n) ~ 5^(n+3/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 17 2016

A128722 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k hills (i.e., peaks at level 1) (0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 22, 9, 4, 0, 1, 84, 35, 12, 5, 0, 1, 334, 138, 49, 15, 6, 0, 1, 1368, 563, 198, 64, 18, 7, 0, 1, 5734, 2352, 825, 264, 80, 21, 8, 0, 1, 24480, 10015, 3504, 1121, 336, 97, 24, 9, 0, 1, 106086, 43308, 15123, 4833, 1452, 414, 115, 27, 10, 0, 1

Offset: 0

Emeric Deutsch, Mar 30 2007

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps.
T(n,0) = A128723(n).
Row sums yield A002212.
Sum_{k=0..n} k*T(n,k) = A033321(n).

			T(3,1)=3 because we have (UD)UUDD, (UD)UUDL and UUDD(UD) (the hills are shown between parentheses).
Triangle starts:
   1;
   0,  1;
   2,  0,  1;
   6,  3,  0,  1;
  22,  9,  4,  0,  1;
  84, 35, 12,  5,  0,  1;

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Cf. A002212, A033321, A128723.

g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-z+z*g)/(1+z-z*g-t*z): Gser:=simplify(series(G,z=0,14)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

G.f.: (1-z+zg)/(1+z-zg-tz), where g = 1+zg^2+z(g-1) = (1-z-sqrt(1-6z+5z^2))/(2z).

cp's OEIS Frontend

A363810 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-14-3, and 4-5-3-1-2.

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A363811 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-3-5-4, and 4-5-3-1-2.

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A363812 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-4-3, and 3-41-2.

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A363813 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-4-3, and 4-5-3-1-2.

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A129167 Number of base pyramids in all skew Dyck paths of semilength n.

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A171224 Riordan array (f(x),x*f(x)) where f(x) is the g.f. of A117641.

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A154930 Inverse of Fibonacci convolution array A154929.

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A171380 Expansion of the first column of triangle T_(1,x), T(x,y) defined in A039599; T_(1,0)= A061554, T_(1,1)= A064189, T_(1,2)= A039599, T_(1,3)= A110877, T_(1,4)= A124576.

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