A061554 -id:A061554 - cp's OEIS frontend

A107430 Triangle read by rows: row n is row n of Pascal's triangle (A007318) sorted into increasing order.

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 6, 1, 1, 5, 5, 10, 10, 1, 1, 6, 6, 15, 15, 20, 1, 1, 7, 7, 21, 21, 35, 35, 1, 1, 8, 8, 28, 28, 56, 56, 70, 1, 1, 9, 9, 36, 36, 84, 84, 126, 126, 1, 1, 10, 10, 45, 45, 120, 120, 210, 210, 252, 1, 1, 11, 11, 55, 55, 165, 165, 330, 330, 462, 462, 1

Offset: 0

Philippe Deléham, May 21 2005

By rows, equals partial sums of A053121 reversed rows. Example: Row 4 of A053121 = (2, 0, 3, 0, 1) -> (1, 0, 3, 0, 2) -> (1, 1, 4, 4, 6). - Gary W. Adamson, Dec 28 2008, edited by Michel Marcus, Sep 22 2015

			Triangle begins:
1;
1,1;
1,1,2;
1,1,3,3;
1,1,4,4,6;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

A061554 is similar but with rows sorted into decreasing order.
Cf. A034868.
Cf. A053121. - Gary W. Adamson, Dec 28 2008
Cf. A103284.

import Data.List (sort)
a107430 n k = a107430_tabl !! n !! k
a107430_row n = a107430_tabl !! n
a107430_tabl = map sort a007318_tabl
-- Reinhard Zumkeller, May 26 2013

/* As triangle */ [[Binomial(n,Floor(k/2)) : k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 22 2015

for n from 0 to 10 do sort([seq(binomial(n,k),k=0..n)]) od; # yields sequence in triangular form. - Emeric Deutsch, May 28 2005

Flatten[ Table[ Sort[ Table[ Binomial[n, k], {k, 0, n}]], {n, 0, 12}]] (* Robert G. Wilson v, May 28 2005 *)

for(n=0,20, for(k=0,n, print1(binomial(n,floor(k/2)), ", "))) \\ G. C. Greubel, May 22 2017

T(n,k) = C(n,floor(k/2)). - Paul Barry, Dec 15 2006; corrected by Philippe Deléham, Mar 15 2007

Sum_{k=0..n} T(n,k)*x^(n-k) = A127363(n), A127362(n), A127361(n), A126869(n), A001405(n), A000079(n), A127358(n), A127359(n), A127360(n) for x=-4,-3,-2,-1,0,1,2,3,4 respectively. - Philippe Deléham, Mar 29 2007

More terms from Emeric Deutsch and Robert G. Wilson v, May 28 2005

A127872 Triangle formed by reading A039599 mod 2.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1

Offset: 0

Philippe Deléham, Apr 05 2007

Also triangle formed by reading triangles A061554, A106180, A110519, A124574, A124576, A126953, A127543 modulo 2.

			Triangle begins:
1;
1, 1;
0, 1, 1;
1, 1, 1, 1;
0, 0, 0, 1, 1;
0, 0, 1, 1, 1, 1;
0, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
0, 0, 0, 0, 0, 0, 0, 1, 1;
0, 0, 0, 0, 0, 0, 1, 1, 1, 1;
0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1;
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1;
0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1;
0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1;
0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A061554, A106180, A110519, A124574, A124576, A126953, A127543.

Cf. A000007, A036987, A001316, A062878, A050614, A000045, A001519

T[0, 0] := 1; T[n_, k_] := Binomial[2*n - 1, n - k] - Binomial[2*n - 1, n - k - 2]; Table[Mod[T[n, k], 2], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Apr 18 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A036987(n), A001316(n), A062878(n) for x=-1,0,1,2 respectively.

Sum_{k=0..n} T(n,k)*Fibonacci(2*k+1) = A050614(n), see A000045 and A001519. - Philippe Deléham, Aug 30 2007

A098493 Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.

Original entry on oeis.org

1, 0, -1, -1, -1, 1, -1, 1, 2, -1, 0, 3, 0, -3, 1, 1, 2, -5, -2, 4, -1, 1, -2, -7, 6, 5, -5, 1, 0, -5, 0, 15, -5, -9, 6, -1, -1, -3, 12, 9, -25, 1, 14, -7, 1, -1, 3, 15, -18, -29, 35, 7, -20, 8, -1, 0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1, 1, 4, -22, -24, 85, 14, -112, 42

Offset: 0

Ralf Stephan, Sep 12 2004

Also, coefficients of polynomials that have values in A098495 and A094954.

			Triangle begins:
   1;
   0, -1;
  -1, -1, 1;
  -1,  1, 2, -1;
   0,  3, 0, -3, 1;
  ...

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.

Columns include A010892, -A076118. Diagonals include A033999, A038608, (-1)^n*A000096. Row sums are in A057077.

Cf. A098494 (diagonal polynomials), A085478, A244419.

A098493 := proc (n, k)
add((-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), j = k..n);
end proc:
seq(seq(A098493(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

T(n,k)=if(k>n||k<0||n<0,0,if(k>=n-1,(-1)^n*if(k==n,1,-k),if(n==1,0,if(k==0,T(n-1,0)-T(n-2,0),T(n-1,k)-T(n-2,k)-T(n-1,k-1)))))

T(n, k) = A098489[n(n+1)/2, k] - A098490[n(n+1)/2, k].
Recurrence: T(n, k) = T(n-1, k)-T(n-1, k-1)-T(n-2, k); T(n, k)=0 for n<0, k>n, k<0; T(n, n)=(-1)^n; T(n, n-1)=(-1)^n*(1-n).
G.f.: (1-x)/(1+(y-1)*x+x^2). [Vladeta Jovovic, Dec 14 2009]
From Peter Bala, Jul 13 2021: (Start)
Riordan array ( (1 - x)/(1 - x + x^2), -x/(1 - x + x^2) ).
T(n,k) = (-1)^k * the (n,k)-th entry of Q^(-1)*P = Sum_{j = k..n} (-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), where P denotes Pascal's triangle A007318 and Q denotes triangle A061554 (formed from P by sorting the rows into descending order). (End)
From Peter Bala, Jun 26 2025: (Start)
n-th row polynomial R(n, x) = Sum_{k = 0..n} (-1)^k * binomial(n+k, 2*k) * (1 + x)^k.
R(n, 2*x + 1) = (-1)^n * Dir(n, x), where Dir(n,x) denotes the n-th row polynomial of the triangle A244419.
R(n, -1 - x) = b(n, x), where b(n, x) denotes the n-th row polynomial of the triangle A085478. (End)

A127358 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*2^(n-k).

Original entry on oeis.org

1, 3, 8, 21, 54, 138, 350, 885, 2230, 5610, 14088, 35346, 88596, 221952, 555738, 1391061, 3480870, 8708610, 21783680, 54483510, 136254964, 340729788, 852000828, 2130354786, 5326563004, 13317759588, 33296999120, 83247698100, 208129274400, 520343244300

Offset: 0

Paul Barry, Jan 11 2007

Hankel transform is (-1)^n. In general, given r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*r^(n-k) has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-2*x) under the Chebyshev mapping g(x) -> (1/sqrt(1-4*x^2))*g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.

			a(3) = 21 = (12 + 6 + 2 + 1), where the top row of M^3 = (12, 6, 2, 1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Cf. A107430. - Philippe Deléham, Sep 16 2009

Table[Sum[Binomial[n,Floor[k/2]]2^(n-k),{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jun 03 2012 *)
CoefficientList[Series[(1 + 2*x - Sqrt[1 - 4*x^2])/(2*Sqrt[1 - 4*x^2]*(x - 1 + Sqrt[1 - 4*x^2])), {x, 0, 50}], x] (* G. C. Greubel, May 22 2017 *)

my(x='x+O('x^50)); Vec((1 + 2*x - sqrt(1 - 4*x^2))/(2*sqrt(1 - 4*x^2)*(x - 1 + sqrt(1 - 4*x^2)))) \\ G. C. Greubel, May 22 2017

G.f.: (1/sqrt(1 - 4x^2))(1 + x*c(x^2))/(1 - 2*x*c(x^2)).
a(n) = 2*a(n-1) + A054341(n-1). a(n) = Sum_{k=0..n} A126075(n,k). - Philippe Deléham, Mar 03 2007
a(n) = Sum_{k=0..n} A061554(n,k)*2^k. - Philippe Deléham, Dec 04 2009
From Gary W. Adamson, Sep 07 2011: (Start)
a(n) is the sum of top row terms of M^n, M is an infinite square production matrix as follows:
  2, 1, 0, 0, 0, ...
  1, 0, 1, 0, 0, ...
  0, 1, 0, 1, 0, ...
  0, 0, 1, 0, 1, ...
  0, 0, 0, 1, 0, ...
  ... (End)
D-finite with recurrence 2*n*a(n) + (-5*n-4)*a(n-1) + 2*(-4*n+13)*a(n-2) + 20*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 30 2012
a(n) ~ 3 * 5^n / 2^(n+1). - Vaclav Kotesovec, Feb 13 2014

A127361 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*(-2)^(n-k).

Original entry on oeis.org

1, -1, 4, -7, 22, -46, 130, -295, 790, -1870, 4864, -11782, 30148, -73984, 187534, -463687, 1168870, -2902870, 7293640, -18161170, 45541492, -113576596, 284470564, -710118262, 1777323772, -4439253196, 11105933440, -27749232700, 69403169200

Offset: 0

Paul Barry, Jan 11 2007

Hankel transform is 3^n. In general, for r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*(-r)^(n-k) has Hankel transform (r+1)^n. The sequence is the image of the sequence with g.f. (1+x)/(1+2*x) under the Chebyshev mapping g(x) -> (1/sqrt(1-4*x^2)) * g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.
Second binomial transform is A026641. - Philippe Deléham, Mar 14 2007
Signed version of A100098. - Philippe Deléham, Nov 25 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A026641, A061554.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1+2*x-Sqrt(1-4*x^2))/(2*Sqrt(1-4*x^2)*(1+x-Sqrt(1-4*x^2))) )); // G. C. Greubel, Feb 17 2019

a:=n->add(binomial(n,floor(k/2))*(-2)^(n-k),k=0..n): seq(a(n),n=0..30); # Muniru A Asiru, Feb 18 2019

CoefficientList[Series[(1/Sqrt[1-4*x^2])*(1+x*(1-Sqrt[1-4*x^2]) / (2*x^2)) /(1+2*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

my(x='x+O('x^30)); Vec( (1+2*x-sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*(1+x-sqrt(1-4*x^2))) ) \\ G. C. Greubel, Feb 17 2019

((1+2*x-sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*(1+x-sqrt(1-4*x^2))) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 17 2019

G.f.: (1/sqrt(1-4*x^2))(1+x*c(x^2))/(1+2*x*c(x^2)), with c(x) = (1 - sqrt(1-4*x))/(2*x).
a(n) = Sum_{k=0..n} A061554(n,k)*(-2)^k. - Philippe Deléham, Nov 25 2007
a(n) = Sum_{k=0..n} A061554(n,k)*(-2)^k. - Philippe Deléham, Dec 04 2009
Conjecture: 2*n*a(n) + (5*n-4)*a(n-1) - 2*(4*n-3)*a(n-2) - 20*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 30 2012
a(n) ~ (-1)^n * 5^n / 2^(n+1). - Vaclav Kotesovec, Feb 13 2014

More terms from Vincenzo Librandi, Feb 15 2014

A127359 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*3^(n-k).

Original entry on oeis.org

1, 4, 14, 48, 162, 544, 1820, 6080, 20290, 67680, 225684, 752448, 2508468, 8362176, 27875064, 92919168, 309734850, 1032458080, 3441543140, 11471842880, 38239537852, 127465249344, 424884399624, 1416281802368, 4720940242612, 15736469278144, 52454901060680

Offset: 0

Paul Barry, Jan 11 2007

Hankel transform is (-2)^n. In general, given r>=0, the sequence given by Sum_{k=0..n} C(n,floor(k/2))*r^(n-k) has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-3x) under the Chebyshev mapping g(x)->(1/sqrt(1-4x^2))*g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Cf. A107430. - Philippe Deléham, Sep 16 2009

Cf. A000108 (Catalan numbers).

a:=[1, 4, 14];; for n in [4..30] do a[n]:=(2*(5*n-2)*a[n-1] +4*(3*n-14)*a[n-2] -40*(n-3)*a[n-3])/(3*(n-1)); od; a; # G. C. Greubel, Dec 15 2019

I:=[1, 4, 14]; [n le 3 select I[n] else (2*(5*n-2)*Self(n-1) + 4*(3*n - 14)*Self(n-2) -40*(n-3)*Self(n-3))/(3*(n-1)): n in [1..30]]; // G. C. Greubel, Dec 15 2019

A127359:=n->sum(binomial(n,floor(k/2))*3^(n-k), k=0..n): seq(A127359(n), n=0..30); # Wesley Ivan Hurt, Mar 14 2015

Table[Sum[Binomial[n, Floor[k/2]]*3^(n-k), {k,0,n}], {n,0,30}] (* Vaclav Kotesovec, Oct 19 2012 *)

a(n) = sum(j=0, n, binomial(n, j\2)*3^(n-j));
vector(31, n, a(n-1)) \\ G. C. Greubel, Dec 15 2019

[sum(binomial(n, floor(j/2))*3^(n-j) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Dec 15 2019

G.f.: (1/sqrt(1-4*x^2))*(1+x*c(x^2))/(1-3*x*c(x^2)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x).
a(n) = Sum_{k=0..n} A061554(n,k)*3^k. - Philippe Deléham, Dec 04 2009
Recurrence: 3*n*a(n) = 2*(5*n + 3)*a(n-1) + 4*(3*n - 11)*a(n-2) - 40*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 4*10^n/3^(n+1). - Vaclav Kotesovec, Oct 19 2012

A158815 Triangle T(n,k) read by rows, matrix product of A046899(row-reversed) * A130595.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 13, 5, 1, 1, 46, 16, 6, 1, 1, 166, 58, 19, 7, 1, 1, 610, 211, 71, 22, 8, 1, 1, 2269, 781, 261, 85, 25, 9, 1, 1, 8518, 2920, 976, 316, 100, 28, 10, 1, 1, 32206, 11006, 3676, 1196, 376, 116, 31, 11, 1, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Mar 27 2009

The left factor of the matrix product is the triangle which starts
   1;
   2,  1;
   6,  3, 1;
  20, 10, 4, 1;
a row-reversed version of A046899, equivalent to the triangular view of the array A092392. The right factor is the inverse of the matrix A007318, which is A130595.
Swapping the two factors, A007318^(-1) * A046899(row-reversed) would generate A158793.
Riordan array (f(x), g(x)) where f(x) is the g.f. of A026641 and where g(x) is the g.f. of A000957. - Philippe Deléham, Dec 05 2009
T(n,k) is the number of nonnegative paths consisting of upsteps U=(1,1) and downsteps D=(1,-1) of length 2n with k low peaks. (A low peak has its peak vertex at height 1.) Example: T(3,1)=5 counts UDUUUU, UDUUUD, UDUUDU, UDUUDD, UUDDUD. - David Callan, Nov 21 2011
Matrix product P^2 * Q * P^(-2), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A171243. - Peter Bala, Jul 13 2021

			The triangle starts
       1;
       1,     1;
       4,     1,     1;
      13,     5,     1,    1;
      46,    16,     6,    1,    1;
     166,    58,    19,    7,    1,   1;
     610,   211,    71,   22,    8,   1,   1;
    2269,   781,   261,   85,   25,   9,   1,  1;
    8518,  2620,   976,  316,  100,  28,  10,  1,  1;
   32206, 11006,  3676, 1196,  376, 116,  31, 11,  1, 1;
  122464, 41746, 13938, 4544, 1442, 441, 133, 34, 12, 1, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Filippo Disanto, Andrea Frosini and Simone Rinaldi, Renzo Pinzani, The Combinatorics of Convex Permutominoes, Southeast Asian Bulletin of Mathematics (2008) 32: 883-912.

Cf. A000984, A026641, A046899, A158793, A171243.

A158815 := proc (n, k)
  add((-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k), j = 0..n);
end proc:
seq(seq(A158815(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

T[n_,k_]:= T[n,k]= Sum[(-1)^(j+k)*Binomial[j,k]*Binomial[2*n-j,n], {j,0,n}];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2021 *)

def A158815(n,k): return sum( (-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k) for j in (0..n) )
flatten([[A158815(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021

Sum_{k=0..n} T(n,k) = A046899(n).
T(n,0) = A026641(n).
Sum_{k=0..n} T(n,k)*x^k = A026641(n), A000984(n), A001700(n), A000302(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Dec 03 2009
T(n, k) = Sum_{j=0..n} binomial(j, k)*binomial(2*n-j, n). - Peter Bala, Jul 13 2021

A127360 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*4^(n-k).

Original entry on oeis.org

1, 5, 22, 95, 406, 1730, 7360, 31295, 133030, 565430, 2403172, 10213670, 43408444, 184486580, 784069252, 3332296895, 14162266630, 60189642830, 255806000260, 1087175537570, 4620496103956, 19637108580380, 83457711731152, 354695275386470, 1507454921406556

Offset: 0

Paul Barry, Jan 11 2007

Hankel transform is (-3)^n. In general, given r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*r^(n-k) has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-4x) under the Chebyshev mapping g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Cf. A107430. - Philippe Deléham, Sep 16 2009

Cf. A061554.

CoefficientList[Series[(1/Sqrt[1-4x^2])*(1+x*(1-Sqrt[1-4*x^2])/(2*x^2))/(1-4*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

G.f.: (1/sqrt(1-4*x^2))*(1+x*c(x^2))/(1-4*x*c(x^2)) with c(x) = (1-sqrt(1-4*x))/(2*x).
a(n) = Sum_{k=0..n} A061554(n,k)*4^k. - Philippe Deléham, Dec 04 2009
Recurrence: 4*n*a(n) = (17*n + 8)*a(n-1) + 2*(8*n - 33)*a(n-2) - 68*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 5*17^n/4^(n+1). - Vaclav Kotesovec, Oct 19 2012

A158793 Triangle read by rows: product of A130595 and A092392 considered as infinite lower triangular arrays.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 19, 9, 5, 1, 1, 51, 26, 11, 6, 1, 1, 141, 70, 34, 13, 7, 1, 1, 393, 197, 92, 43, 15, 8, 1, 1, 1107, 553, 265, 117, 53, 17, 9, 1, 1, 3139, 1570, 751, 346, 145, 64, 19, 10, 1, 1, 8953, 4476, 2156, 991, 441, 176, 76, 21, 11, 1, 1

Offset: 0

Gary W. Adamson & Roger L. Bagula, Mar 26 2009

Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A002426 and where g(x) is the g.f. of A005043. - Philippe Deléham, Dec 05 2009

Matrix product P * Q * P^(-1), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158815 and A171243. - Peter Bala, Jul 13 2021

			First rows of the triangle:
     1;
     1,    1;
     3,    1,    1;
     7,    4,    1,   1;
    19,    9,    5,   1,   1;
    51,   26,   11,   6,   1,   1;
   141,   70,   34,  13,   7,   1,  1;
   393,  197,   92,  43,  15,   8,  1,  1;
  1107,  553,  265, 117,  53,  17,  9,  1,  1;
  3139, 1570,  751, 346, 145,  64, 19, 10,  1, 1;
  8953, 4476, 2156, 991, 441, 176, 76, 21, 11, 1, 1;

T(n, 0) = A002426(n), A005773 (row sums).

Cf. A046899, A007318, A158815, A171243.

A158793 := proc (n, k)
  add((-1)^(n+j)*binomial(n, j)*binomial(2*j-k, j-k), j = k..n);
end proc:
seq(seq(A158793(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

T[n_, k_] := (-1)^(k + n) Binomial[n, k] HypergeometricPFQ[{k/2 + 1/2, k/2 + 1, k - n}, {k + 1, k + 1}, 4];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Jul 17 2021 *)

T(n, m) = Sum_{k=m..n-1} A130595(n,k) * A092392(k+1,m+1), with the triangular interpretation of A092392.
Conjecture: T(n,1) = A113682(n-1). - R. J. Mathar, Oct 06 2009
Sum_{k=0..n} T(n,k)*x^k = A002426(n), A005773(n+1), A000244(n), A126932(n) for x = 0,1,2,3 respectively. - Philippe Deléham, Dec 03 2009
T(n, k) = (-1)^(k + n) binomial(n, k) hypergeom([k/2 + 1/2, k/2 + 1, k - n], [k + 1, k + 1], 4). - Peter Luschny, Jul 17 2021

Simplified definition from R. J. Mathar, Oct 06 2009

A110438 Triangular array giving the number of NSEW unit step lattice paths of length n with terminal height k subject to the following restrictions. The paths start at the origin (0,0) and take unit steps (0,1)=N(north), (0,-1)=S(south), (1,0)=E(east) and (-1,0)=W(west) such that no paths pass below the x-axis, no paths begin with W, all W steps remain on the x-axis and there are no NS steps.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 12, 10, 7, 4, 1, 29, 25, 18, 11, 5, 1, 71, 62, 47, 30, 16, 6, 1, 175, 155, 121, 82, 47, 22, 7, 1, 434, 389, 311, 220, 135, 70, 29, 8, 1, 1082, 979, 799, 584, 378, 212, 100, 37, 9, 1, 2709, 2471, 2051, 1541, 1039, 620, 320, 138, 46, 10, 1

Offset: 0

Asamoah Nkwanta (Nkwanta(AT)jewel.morgan.edu), Aug 10 2005

The row sums are the even-indexed Fibonacci numbers.

Matrix product Q^(-1) * P * Q, where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793. - Peter Bala, Jul 14 2021

			Triangle starts:
  1;
  1,1;
  2,2,1;
  5,4,3,1;
  12,10,7,4,1;

A. Nkwanta, A Riordan matrix approach to unifying a selected class of combinatorial arrays, Congressus Numerantium, 160 (2003), pp. 33-55.
A. Nkwanta, A note on Riordan matrices, Contemporary Mathematics Series, AMS, 252 (1999), pp. 99-107.
A. Nkwanta, Lattice paths, generating functions and the Riordan group, Ph.D. Thesis, Howard University, Washington DC, 1997.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
Tian-Xiao He, A-sequences, Z-sequence, and B-sequences of Riordan matrices, Discrete Mathematics 343.3 (2020): 111718.

Row sums are A001519(n+1).

Cf. A097724, A158793.

A110438 := proc (n, k)
    add((-1)^binomial(n-i+1, 2)*binomial(floor((1/2)*n+(1/2)*i), i)*add(binomial(i, j)*binomial(j, floor((1/2)*j-(1/2)*k)), j = k..i), i = 0..n);
end proc:
seq(seq(A110438(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 14 2021

\\ ColGf gives g.f. of k-th column.
ColGf(k,n)={my(g=(1 - x + x^2 - sqrt(1 - 2*x - x^2 - 2*x^3 + x^4 + O(x^(n-k+3))))/(2*x^2)); (1 - x)*g/(1 - x*g)*(x*g)^k}
T(n,k) = {polcoef(ColGf(k,n), n)} \\ Andrew Howroyd, Mar 02 2023

Recurrence is d(0, 0) = 1, d(1, 0) = 1, d(n+1, 0) = 2*d(n, 0) + Sum_{j>=1} d(n-j, j), n>=1 for leftmost column and d(n+1, k) = d(n, k-1) + d(n, k) + Sum_{j>=1} d(n-j, k+j), n>=2, k>=1 and n>j; Riordan array d(n, k): (((1-z)/(2*z))*(sqrt(1+z+z^2)/sqrt(1-3*z+z^2) - 1), ((1-z+z^2)-sqrt(1-2*z-z^2-2*z^3+z^4))/(2*z)).

Terms a(55) and beyond from Andrew Howroyd, Mar 02 2023

cp's OEIS Frontend

A107430 Triangle read by rows: row n is row n of Pascal's triangle (A007318) sorted into increasing order.

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A127872 Triangle formed by reading A039599 mod 2.

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A098493 Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.

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A127358 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*2^(n-k).

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A127361 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*(-2)^(n-k).

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A127359 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*3^(n-k).

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