A053121 -id:A053121 - cp's OEIS frontend

1, 3, 1, 8, 4, 1, 20, 12, 5, 1, 48, 32, 17, 6, 1, 112, 80, 49, 23, 7, 1, 256, 192, 129, 72, 30, 8, 1, 576, 448, 321, 201, 102, 38, 9, 1, 1280, 1024, 769, 522, 303, 140, 47, 10, 1, 2816, 2304, 1793, 1291, 825, 443, 187, 57, 11, 1, 6144, 5120, 4097, 3084, 2116, 1268, 630

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((1-z)/(1-2*z)^2)/(1-x*z/(1-z)).
This is the second member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
The column sequences appear in A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 for m=0..7.

			1;
3,1;
8,4,1;
20,12,5,1;
...
Fourth row polynomial (n=3): p(3,x)= 20+12*x+5*x^2+x^3

G. C. Greubel, Table of n, a(n) for n = 0..1274

Cf. A007318, A055248, A008949. Row sums: A049611(n+1) = A055252(n, 0).

a[n_, m_] := Binomial[n, m]*Hypergeometric2F1[2, m-n, m+1, -1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 11 2014 *)

a(n, m) = Sum_{k=m,..,n} ( A055248(n, k) ), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m) = Sum_{j=m,..,(n-1)} ( a(j, m) ) + A055248(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: ((1-x)/(1-2*x)^2)*(x/(1-x))^m, m >= 0.

a(n, m) = binomial(n, m) * 2F1(2, m-n; m+1; -1) where 2F1 is the hypergeometric function. Jean-François Alcover, Mar 11 2014

A093564 (7,1) Pascal triangle.

Original entry on oeis.org

1, 7, 1, 7, 8, 1, 7, 15, 9, 1, 7, 22, 24, 10, 1, 7, 29, 46, 34, 11, 1, 7, 36, 75, 80, 45, 12, 1, 7, 43, 111, 155, 125, 57, 13, 1, 7, 50, 154, 266, 280, 182, 70, 14, 1, 7, 57, 204, 420, 546, 462, 252, 84, 15, 1, 7, 64, 261, 624, 966, 1008, 714, 336, 99, 16, 1, 7, 71, 325, 885

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(7;n,m) gives in the columns m>=1 the figurate numbers based on A016993, including the 9-gonal numbers A001106, (see the W. Lang link).
This is the seventh member, d=7, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-3, for d=1..6.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=(1+6*z)/(1-(1+x)*z).
The SW-NE diagonals give A022097(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 6. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

			Triangle begins
  [1];
  [7,  1];
  [7,  8,  1];
  [7, 15,  9,  1];
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch. 5, pp. 109-122.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
W. Lang, First 10 rows and array of figurate numbers .

Row sums: A000079(n+2), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 6 for n=2 and 0 otherwise.
The column sequences give for m=1..9: A016993, A001106 (9-gonal), A007584, A051740, A051877, A050403, A027818, A034266, A055994.
Cf. A093565 (d=8).

a093564 n k = a093564_tabl !! n !! k
a093564_row n = a093564_tabl !! n
a093564_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [7, 1]
-- Reinhard Zumkeller, Sep 01 2014

N:= 20: # to get the first N rows
T:=Matrix(N,N):
T[1,1]:= 1:
for m from 2 to N do
T[m,1]:= 7:
T[m,2..m]:= T[m-1,1..m-1] + T[m-1,2..m];
od:
for m from 1 to N do
convert(T[m,1..m],list)
od; # Robert Israel, Dec 28 2014

a(n, m)=F(7;n-m, m) for 0<= m <= n, otherwise 0, with F(7;0, 0)=1, F(7;n, 0)=7 if n>=1 and F(7;n, m):=(7*n+m)*binomial(n+m-1, m-1)/m if m>=1.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=7 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+6*x)/(1-x)^(m+1), m>=0.
T(n, k) = C(n, k) + 6*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(7 + 15*x + 9*x^2/2! + x^3/3!) = 7 + 22*x + 46*x^2/2! + 80*x^3/3! + 125*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A189391 The minimum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 0 to n, and each number in a higher row is the sum of the two numbers directly below it.

Original entry on oeis.org

0, 1, 3, 8, 19, 44, 98, 216, 467, 1004, 2134, 4520, 9502, 19928, 41572, 86576, 179587, 372044, 768398, 1585416, 3263210, 6711176, 13775068, 28255568, 57863214, 118430584, 242061468, 494523536, 1009105372, 2058327344, 4194213448

Offset: 0

Nathaniel Johnston, Apr 20 2011

This is the Riordan transform of A000217 (triangular numbers) with the Riordan matrix (of the Bell type) A053121 (inverse of the Chebyshev S Bell matrix). See the resulting formulae below. - Wolfdieter Lang, Feb 18 2017.

Conjecture: a(n) is also half the sum of the "cuts-resistance" (see A319416, A319420, A319421) of all binary vectors of length n (see Lenormand, page 4). - N. J. A. Sloane, Sep 20 2018

			For n = 4 consider the triangle:
....19
...8  11
..5  3  8
.4  1 2  6
3  1 0 2  4
This triangle has 19 at its apex and no other such triangle with the numbers 0 - 4 on its base has a smaller apex value, so a(4) = 19.

Nathaniel Johnston, Table of n, a(n) for n = 0..1000
F. Disanto and S. Rinaldi, Symmetric convex permutominoes and involutions, PU. M. A., Vol. 22 (2011), No. 1, pp. 39-60. - From _N. J. A. Sloane_, May 04 2012
Steven R. Finch, How far might we walk at random?, arXiv:1802.04615 [math.HO], 2018.
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - _N. J. A. Sloane_, Sep 20 2018

Cf. A066411, A099325, A189390, A189162, A000217, A053121, A281862.

Cf. also A319416, A319420, A319421.

m:=30; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!((2*x+Sqrt(1-4*x^2)-1)/(2*(2*x-1)^2))); // G. C. Greubel, Aug 24 2018

a:=proc(n)return add((2*n-4*k-1)*binomial(n,k),k=0..floor((n-1)/2)): end:
seq(a(n),n=0..50);

CoefficientList[Series[(2*x+Sqrt[1-4*x^2]-1) / (2*(2*x-1)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 16 2014 *)

A189391(n)=sum(i=0,(n-1)\2,(2*n-4*i-1)*binomial(n,i))  \\ M. F. Hasler, Jan 24 2012

If n even, a(n) = (n+1/2)*binomial(n,n/2) - 2^(n-1); if n odd, a(n) = ((n+1)/2)*binomial(n+1,(n+1)/2) - 2^(n-1). - N. J. A. Sloane, Nov 01 2018
a(n) = Sum_{k=0..floor((n-1)/2)} (2*n-4*k-1)*binomial(n,k).
G.f.: (2*x+sqrt(1-4*x^2)-1) / (2*(2*x-1)^2). - Alois P. Heinz, Feb 09 2012
a(n) ~ 2^n * (sqrt(2n/Pi)- 1/2). - Vaclav Kotesovec, Mar 16 2014 (formula simplified by Lewis Chen, May 25 2017)
D-finite with recurrence n*a(n) + (n-5)*a(n-1) + 2*(-5*n+6)*a(n-2) + 4*(-n+8)*a(n-3) + 24*(n-3)*a(n-4) = 0. - R. J. Mathar, Jan 04 2017
From Wolfdieter Lang, Feb 18 2017:(Start)
a(n) = Sum_{m=0..n} A053121(n, m)*A000217(m), n >= 0.
G.f.: c(x^2)*Tri(x*c(x^2)), with c and Tri the g.f. of A000108 and A000217, respectively. See the explicit form of the g.f. given above by Alois P. Heinz.
(End)
2*a(n) = A152548(n)-2^n. - R. J. Mathar, Jun 17 2021

A054336 A convolution triangle of numbers based on A001405 (central binomial coefficients).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 6, 10, 9, 4, 1, 10, 22, 22, 14, 5, 1, 20, 44, 54, 40, 20, 6, 1, 35, 93, 123, 109, 65, 27, 7, 1, 70, 186, 281, 276, 195, 98, 35, 8, 1, 126, 386, 618, 682, 541, 321, 140, 44, 9, 1, 252, 772, 1362, 1624, 1440, 966, 497, 192, 54, 10, 1

Offset: 0

Wolfdieter Lang, Mar 13 2000

T(n,k) is the number of 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) with no level steps at positive height and having k blue level steps. Example: T(4,2)=9 because, denoting U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0), we have BBRR, BRBR, BRRB, RBBR, RBRB, RRBB, BBUD, BUDB, and UDBB. - Emeric Deutsch, Jun 07 2011
In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/(1-(1+x)*z-z^2*c(z^2)), with c(x) the g.f. for Catalan numbers A000108.
Column sequences: A001405, A045621.
Riordan array (f(x), x*f(x)), f(x) the g.f. of A001405. - Philippe Deléham, Dec 08 2009
From Paul Barry, Oct 21 2010: (Start)
Riordan array ((sqrt(1+2x) - sqrt(1-2x))/(2x*sqrt(1-2x)), (sqrt(1+2x)-sqrt(1-2x))/(2*sqrt(1-2x))),
inverse of Riordan array ((1+x)/(1+2x+2x^2), x(1+x)/(1+2x+2x^2)) (A181472). (End)

			Fourth row polynomial (n=3): p(3,x)= 3 + 5*x + 3*x^2 + x^3.
From _Paul Barry_, Oct 21 2010: (Start)
Triangle begins
   1;
   1,  1;
   2,  2,   1;
   3,  5,   3,   1;
   6, 10,   9,   4,  1;
  10, 22,  22,  14,  5,  1;
  20, 44,  54,  40, 20,  6, 1;
  35, 93, 123, 109, 65, 27, 7, 1;
Production matrix is
   1,  1;
   1,  1,  1;
  -1,  1,  1,  1;
   1, -1,  1,  1,  1;
  -1,  1, -1,  1,  1,  1;
   1, -1,  1, -1,  1,  1,  1;
  -1,  1, -1,  1, -1,  1,  1, 1;
   1, -1,  1, -1,  1, -1,  1, 1, 1;
  -1,  1, -1,  1, -1,  1, -1, 1, 1, 1; (End)

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

Cf. A001405, A035324, A054335.

Row sums: A054341.

A053121:= function(n,k)
    if ((n-k+1) mod 2)=0 then return 0;
    else return (k+1)*Binomial(n+1, Int((n-k)/2))/(n+1);
    fi;
  end;
T:= function(n,k)
    return Sum([k..n], j-> Binomial(j,k)*A053121(n,j));
  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jul 21 2019

A053121:= func< n,k | ((n-k+1) mod 2) eq 0 select 0 else (k+1)*Binomial(n+1, Floor((n-k)/2))/(n+1) >;
T:= func< n,k | (&+[Binomial(j,k)*A053121(n,j): j in [k..n]]) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 21 2019

c[n_, j_] /; n < j || OddQ[n - j] = 0; c[n_, j_] = (j + 1) Binomial[n + 1, (n - j)/2]/(n + 1); t[n_, k_] := Sum[c[n, j]*Binomial[j, k], {j, 0, n}]; Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]][[;; 66]] (* Jean-François Alcover, Jul 13 2011, after Philippe Deléham *)

A053121(n,k) = if((n-k+1)%2==0, 0, (k+1)*binomial(n+1, (n-k)\2)/(n+1) );
T(n,k) = sum(j=k,n, A053121(n,j)*binomial(j,k));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 21 2019

def A053121(n, k):
    if (n-k+1) % 2==0: return 0
    else: return (k+1)*binomial(n+1, ((n-k)//2))/(n+1)
def T(n,k): return sum(binomial(j,k)*A053121(n,j) for j in (k..n))
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 21 2019

G.f. for column m: cbi(x)*(x*cbi(x))^m, with cbi(x) := (1+x*c(x^2))/sqrt(1-4*x^2) = 1/(1-x-x^2*c(x^2)), where c(x) is the g.f. for Catalan numbers A000108.
T(n,k) = Sum_{j>=0} A053121(n,j)*binomial(j,k). - Philippe Deléham, Mar 30 2007
T(n,k) = T(n-1,k-1) + T(n-1,l) + Sum_{j>=0} T(n-1,k+1+j)*(-1)^j. - Philippe Deléham, Feb 23 2012

A054341 Row sums of triangle A054336 (central binomial convolutions).

Original entry on oeis.org

1, 2, 5, 12, 30, 74, 185, 460, 1150, 2868, 7170, 17904, 44760, 111834, 279585, 698748, 1746870, 4366460, 10916150, 27287944, 68219860, 170541252, 426353130, 1065853432, 2664633580, 6661479944, 16653699860, 41633878200, 104084695500, 260210401530, 650526003825

Offset: 0

Wolfdieter Lang, Mar 13 2000

a(n) = # Dyck (n+1)-paths all of whose components are symmetric. A strict Dyck path is one with exactly one return to ground level (necessarily at the end). Every nonempty Dyck path is expressible uniquely as a concatenation of one or more strict Dyck paths, called its components. - David Callan, Mar 02 2005
a(n) = # 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) with no level steps at positive height. Example: a(2)=5 because, denoting U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0), we have BB, BR, RB, RR, and UD. - Emeric Deutsch, Jun 07 2011
Inverse Chebyshev transform of the second kind applied to 2^n. This maps g(x) -> c(x^2)g(xc(x^2)). - Paul Barry, Sep 14 2005
Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007
Inverse binomial transform of A059738. - Philippe Deléham, Nov 24 2009

			a(4) = 30, the upper left term of M^4.

Alois P. Heinz, Table of n, a(n) for n = 0..2513 (first 201 terms from Vincenzo Librandi)
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A000108, A054336.

b:= proc(x, y) option remember; `if`(x=0, 1,
      b(x-1, 0)+`if`(y>0, b(x-1, y-1), 0)+b(x-1, y+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..31);  # Alois P. Heinz, Jan 23 2024

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

a(n) = Sum_{m=0..n} A054336(n, m).
G.f.: 1/(1-2*x-x^2*c(x^2)), where c(x) = g.f. for Catalan numbers A000108.
From_Paul Barry_, Sep 14 2005: (Start)
G.f.: c(x^2)/(1-2*x*c(x^2));
a(n) = Sum_{k=0..n} binomial(n,(n-k)/2)*(1 + (-1)^(n+k))*2^k*(k+1)/(n+k+2). (End)
G.f.: 2/(1-4*x+sqrt(1-4*x^2)). - Ira M. Gessel, Oct 27 2013
a(n) = A127358(n+1) - 2*A127358(n). - Philippe Deléham, Mar 02 2007
a(n) = A126075(n,0). - Philippe Deléham, Nov 24 2009
a(n) = Sum_{k=0..n} A053121(n,k)*2^k. - Philippe Deléham, Nov 28 2009
From Gary W. Adamson, Sep 07 2011: (Start)
a(n) is the upper left term of M^n, M is an infinite square production matrix as follows:
  2, 1, 0, 0, 0, ...
  1, 0, 2, 0, 0, ...
  0, 1, 0, 1, 0, ...
  0, 0, 1, 0, 1, ...
  0, 0, 0, 1, 0, ...
  ... (End)
Conjecture: 2*(n+1)*a(n) +5*(-n-1)*a(n-1) +8*(-n+2)*a(n-2) +20*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 30 2012
a(n) ~ 3 * 5^n / 2^(n+2). - Vaclav Kotesovec, Feb 13 2014

A096956 Pascal (1,6) triangle.

Original entry on oeis.org

6, 1, 6, 1, 7, 6, 1, 8, 13, 6, 1, 9, 21, 19, 6, 1, 10, 30, 40, 25, 6, 1, 11, 40, 70, 65, 31, 6, 1, 12, 51, 110, 135, 96, 37, 6, 1, 13, 63, 161, 245, 231, 133, 43, 6, 1, 14, 76, 224, 406, 476, 364, 176, 49, 6, 1, 15, 90, 300, 630, 882, 840, 540, 225, 55, 6, 1, 16, 105, 390, 930

Offset: 0

Wolfdieter Lang, Aug 13 2004

Except for the first row this is the row reversed (6,1)-Pascal triangle A093563.
This is the sixth member, q=6, in the family of (1,q) Pascal triangles: A007318 (Pascal (q=1)), A029635 (q=2) (but with a(0,0)=2, not 1), A095660 (q=3), A095666 (q=4), A096940 (q=5).
This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column no. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(6-5*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(6-5*z)/(1-(1+x)*z).
The SW-NE diagonals give Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k) = A022097(n-2), n >= 2, with n=1 value 6. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

			Triangle begins:
  [0]  6;
  [1]  1,  6;
  [2]  1,  7,  6;
  [3]  1,  8, 13,  6;
  [4]  1,  9, 21, 19,  6;
  [5]  1, 10, 30, 40, 25,  6;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Wolfdieter Lang, First 10 rows.

Row sums: A005009(n-1), n>=1, 6 if n=0; g.f.: (6-5*x)/(1-2*x). Alternating row sums are [6, -5, followed by 0's].

Column sequences (without leading zeros) give for m=1..9, with n >= 0: A000027(n+6), A056115, A096957-9, A097297-A097300.

a(n,k):=piecewise(n=0,6,0Mircea Merca, Apr 08 2012

A096956[n_, k_] := If[n == k, 6, (5*k/n + 1)*Binomial[n, k]];
Table[A096956[n, k], {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Apr 14 2025 *)

Recursion: a(n,m)=0 if m > n, a(0,0) = 6; a(n,0) = 1 if n >= 1; a(n,m) = a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (6-5*x)/(1-x)^(m+1), m >= 0.
a(n,k) = (1+5*k/n)*binomial(n,k), for n > 0. - Mircea Merca, Apr 08 2012

A008313 Triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 1, 5, 4, 1, 5, 9, 5, 1, 14, 14, 6, 1, 14, 28, 20, 7, 1, 42, 48, 27, 8, 1, 42, 90, 75, 35, 9, 1, 132, 165, 110, 44, 10, 1, 132, 297, 275, 154, 54, 11, 1, 429, 572, 429, 208, 65, 12, 1, 429, 1001, 1001, 637, 273, 77, 13, 1

Offset: 0

N. J. A. Sloane

This is another reading (by shallow diagonals) of the triangle A009766; rows of Catalan triangle A008315 read backwards. - Philippe Deléham, Feb 15 2004

"The Catalan triangle is formed in the same manner as Pascal's triangle, except that no number may appear on the left of the vertical bar." [Conway and Smith]

			.|...1
.|.......1
.|...1.......1
.|.......2.......1
.|...2.......3.......1
.|.......5.......4.......1
.|...5.......9.......5.......1
.|......14......14.......6.......1
.|..14......28......20.......7.......1
.|......42......48......27.......8.......1

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)
P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 30, Nos. 3/4, 1994, 52-54.

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
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].
I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838 [math.CO], 2015-2018.
Tom Halverson, Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.
Vaughan F. R. Jones, The Jones Polynomial, 18 August 2005, see the diagram on page 7. - Paul Curtz, Jun 22 2011
P. Mongelli, Kazhdan-Lusztig polynomials of Boolean elements, arXiv preprint arXiv:1111.2945 [math.CO], 2011.
Index entries for sequences related to Chebyshev polynomials.

Cf. A039598, A039599. A053121 is essentially the same triangle.

Row sums = A001405 (central binomial coefficients).

a008313 n k = a008313_tabf !! n !! k
a008313_row n = a008313_tabf !! n
a008313_tabf = map (filter (> 0)) a053121_tabl
-- Reinhard Zumkeller, Feb 24 2012

T := proc(n, k): if n=0 then 1 else binomial(n-1, floor(n/2 )-k) -binomial(n-1, floor(n/2) -k-2) fi: end: seq(seq(T(n, k), k = 0..floor(n/2)), n = 0..14); # Johannes W. Meijer, Jul 10 2011, revised Nov 22 2012

t[n_, k_] /; n < k || OddQ[n - k] = 0; t[n_, k_] := (k+1)*Binomial[n+1, (n-k)/2]/(n+1); Flatten[ Table[ t[n, k], {n, 0, 15}, {k, Mod[n, 2], n + Mod[n, 2], 2}]] (* Jean-François Alcover, Jan 12 2012 *)

{T(n, k) = if( k<0 || 2*k>n, 0, polcoeff((1 - x) * (1 + x)^n, n\2 - k))}; /* Michael Somos, May 28 2005 */

T(n, k) = binomial(n-1, n\2-k)-binomial(n-1, n\2-k-2);
for(n=0, 14, for(k=0, n\2, print1(T(n,k),", "))); \\ Seiichi Manyama, Mar 24 2025

# Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle.
def A008313_triangle(n) :
    D = [0]*((n+5)//2); D[1] = 1
    b = True; h = 1
    for i in range(n) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        b = not b
        print([D[z] for z in (1..h-1)])
A008313_triangle(13) # Peter Luschny, May 01 2012

Row n: C(n-1, [n/2]-k) - C(n-1, [n/2]-k-2) for k=0, 1, ..., n.

Sum_{k>=0} T(n, k)^2 = A000108(n); A000108: Catalan numbers. - Philippe Deléham, Feb 14 2004

A054335 A convolution triangle of numbers based on A000984 (central binomial coefficients of even order).

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 20, 16, 6, 1, 70, 64, 30, 8, 1, 252, 256, 140, 48, 10, 1, 924, 1024, 630, 256, 70, 12, 1, 3432, 4096, 2772, 1280, 420, 96, 14, 1, 12870, 16384, 12012, 6144, 2310, 640, 126, 16, 1, 48620, 65536, 51480, 28672, 12012, 3840, 924, 160, 18, 1

Offset: 0

Wolfdieter Lang, Mar 13 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/(sqrt(1-4*z)-x*z).
The column sequences are for m=0..20: A000984, A000302 (powers of 4), A002457, A002697, A002802, A038845, A020918, A038846, A020920, A040075, A020922, A045543, A020924, A054337, A020926, A054338, A020928, A054339, A020930, A054340, A020932.
Riordan array (1/sqrt(1-4*x),x/sqrt(1-4*x)). - Paul Barry, May 06 2009
The matrix inverse is apparently given by deleting the leftmost column from A206022. - R. J. Mathar, Mar 12 2013

			Triangle begins:
    1;
    2,    1;
    6,    4,   1;
   20,   16,   6,   1;
   70,   64,  30,   8,  1;
  252,  256, 140,  48, 10,  1;
  924, 1024, 630, 256, 70, 12, 1; ...
Fourth row polynomial (n=3): p(3,x) = 20 + 16*x + 6*x^2 + x^3.
From _Paul Barry_, May 06 2009: (Start)
Production matrix begins
    2,   1;
    2,   2,  1;
    0,   2,  2,  1;
   -2,   0,  2,  2,  1;
    0,  -2,  0,  2,  2,  1;
    4,   0, -2,  0,  2,  2, 1;
    0,   4,  0, -2,  0,  2, 2, 1;
  -10,   0,  4,  0, -2,  0, 2, 2, 1;
    0, -10,  0,  4,  0, -2, 0, 2, 2, 1; (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A000984, A035324, A054336.

Row sums: A026671.

T:= function(n, k)
    if k mod 2=0 then return Binomial(2*n-k, n-Int(k/2))*Binomial(n-Int(k/2),Int(k/2))/Binomial(k,Int(k/2));
    else return 4^(n-k)*Binomial(n-Int((k-1)/2)-1, Int((k-1)/2));
    fi;
  end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Jul 20 2019

T:= func< n, k | (k mod 2) eq 0 select Binomial(2*n-k, n-Floor(k/2))* Binomial(n-Floor(k/2),Floor(k/2))/Binomial(k,Floor(k/2)) else 4^(n-k)*Binomial(n-Floor((k-1)/2)-1, Floor((k-1)/2)) >;
[[T(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jul 20 2019

A054335 := proc(n,k)
    if k <0 or k > n then
        0 ;
    elif type(k,odd) then
        kprime := floor(k/2) ;
        binomial(n-kprime-1,kprime)*4^(n-k) ;
    else
        kprime := k/2 ;
        binomial(2*n-k,n-kprime)*binomial(n-kprime,kprime)/binomial(k,kprime) ;
    end if;
end proc: # R. J. Mathar, Mar 12 2013
# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> binomial(2*(n-1), n-1)); # Peter Luschny, Oct 19 2022

Flatten[ CoefficientList[#1, x] & /@ CoefficientList[ Series[1/(Sqrt[1 - 4*z] - x*z), {z, 0, 9}], z]] (* or *)
a[n_, k_?OddQ] := 4^(n-k)*Binomial[(2*n-k-1)/2, (k-1)/2]; a[n_, k_?EvenQ] := (Binomial[n-k/2, k/2]*Binomial[2*n-k, n-k/2])/Binomial[k, k/2]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 08 2011, updated Jan 16 2014 *)

T(n, k) = if(k%2==0, binomial(2*n-k, n-k/2)*binomial(n-k/2,k/2)/binomial(k,k/2), 4^(n-k)*binomial(n-(k-1)/2-1, (k-1)/2));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 20 2019

def T(n, k):
    if (mod(k,2)==0): return binomial(2*n-k, n-k/2)*binomial(n-k/2,k/2)/binomial(k,k/2)
    else: return 4^(n-k)*binomial(n-(k-1)/2-1, (k-1)/2)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 20 2019

a(n, 2*k+1) = binomial(n-k-1, k)*4^(n-2*k-1), a(n, 2*k) = binomial(2*(n-k), n-k)*binomial(n-k, k)/binomial(2*k, k), k >= 0, n >= m >= 0; a(n, m) := 0 if n

Column recursion: a(n, m)=2*(2*n-m-1)*a(n-1, m)/(n-m), n>m >= 0, a(m, m) := 1.

G.f. for column m: cbie(x)*(x*cbie(x))^m, with cbie(x) := 1/sqrt(1-4*x).

G.f.: 1/(1-x*y-2*x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction). - Paul Barry, May 06 2009

Sum_{k>=0} T(n,2*k)*(-1)^k*A000108(k) = A000108(n+1). - Philippe Deléham, Jan 30 2012

Sum_{k=0..floor(n/2)} T(n-k,n-2*k) = A098615(n). - Philippe Deléham, Feb 01 2012

T(n,k) = 4*T(n-1,k) + T(n-2,k-2) for k>=1. - Philippe Deléham, Feb 02 2012

Vertical recurrence: T(n,k) = 1*T(n-1,k-1) + 2*T(n-2,k-1) + 6*T(n-3,k-1) + 20*T(n-4,k-1) + ... for k >= 1 (the coefficients 1, 2, 6, 20, ... are the central binomial coefficients A000984). - Peter Bala, Oct 17 2015

A097609 Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k horizontal steps at level 0.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 2, 3, 0, 1, 6, 7, 3, 4, 0, 1, 15, 14, 12, 4, 5, 0, 1, 36, 37, 24, 18, 5, 6, 0, 1, 91, 90, 67, 36, 25, 6, 7, 0, 1, 232, 233, 165, 106, 50, 33, 7, 8, 0, 1, 603, 602, 438, 264, 155, 66, 42, 8, 9, 0, 1, 1585, 1586, 1147, 719, 390, 215, 84, 52, 9, 10, 0, 1

Offset: 0

Emeric Deutsch, Aug 30 2004

Row sums give the Motzkin numbers (A001006).
Column 0 is A005043.
Riordan array ((1+x-sqrt(1-2*x-3*x^2))/(2*x*(1-x)), (1+x-sqrt(1-2*x-3*x^2))/(2*(1-x))). - Paul Barry, Jun 21 2008
Inverse of Riordan array ((1-x)/(1-x+x^2), x*(1-x)/(1-x+x^2)), which is A104597. - Paul Barry, Jun 21 2008
Triangle read by rows, product of A064189 and A130595 considered as infinite lower triangular arrays; A097609 = A064189*A130195 = B*A053121*B^(-1) where B = A007318. - Philippe Deléham, Dec 07 2009
T(n+1,1) = A187306(n). - Philippe Deléham, Jan 28 2014
The number of lattice paths from (0,0) to (n,k) that do not cross below the x-axis and use up-step=(1,1) and down-steps=(1,-z) where z is a positive integer. For example, T(4,0) = 3: [(1,1)(1,1)(1,-1)(1,-1)], [(1,1)(1,-1)(1,1)(1,-1)] and [(1,1)(1,1)(1,1)(1,-3)]. - Nicholas Ham, Aug 20 2015
The convolution triangle of the Riordan numbers A005043. - Peter Luschny, Oct 09 2022

			Triangle begins:
  1;
  0, 1;
  1, 0, 1;
  1, 2, 0, 1;
  3, 2, 3, 0, 1;
  6, 7, 3, 4, 0, 1;
Row n has n+1 terms.
T(5,2) = 3 because (HH)UHD,(H)UHD(H) and UHD(HH) are the only Motzkin paths of length 5 with 2 horizontal steps at level 0 (shown between parentheses); here U=(1,1), H=(1,0) and D=(1,-1).
Production matrix begins
  0, 1;
  1, 0, 1;
  1, 1, 0, 1;
  1, 1, 1, 0, 1;
  1, 1, 1, 1, 0, 1;
  1, 1, 1, 1, 1, 0, 1;
  1, 1, 1, 1, 1, 1, 0, 1;
  1, 1, 1, 1, 1, 1, 1, 0, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 0, 1;
... - _Philippe Deléham_, Mar 02 2013

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
Igor Dolinka, James East, and Robert D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015.
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Trends in Mathematics 2000, pp 127-139.

Cf. A001006, A005043, A027907, A187306.

[((k+1)/(n+1))*(&+[(-1)^(n-j+1)*Binomial(n+1,j)*Binomial(2*j-k-2,j-1): j in [k+1..n+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

G:=2/(1-2*t*z+z+sqrt(1-2*z-3*z^2)): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: seq(seq(coeff(t*P[n], t^k),k=1..n+1),n=0..12);
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> A005043(n-1)); # Peter Luschny, Oct 09 2022

nmax = 12; t[n_, k_] := ((-1)^(n+k)*k*n!*HypergeometricPFQ[{(k+1)/2, k/2, k-n}, {k, k+1}, 4])/(n*k!*(n-k)!); Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)

T(n,k) = ((k+1)/(n+1))*sum(j=k+1, n+1, (-1)^(n-j+1)*binomial(n+1,j)* binomial(2*j-k-2,j-1) ); \\ G. C. Greubel, Feb 18 2020

[[((k+1)/(n+1))*sum( (-1)^(n-j+1)*binomial(n+1,j)* binomial(2*j-k-2,j-1) for j in (k+1..n+1)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 18 2020

G.f.: 2/(1 -2*t*z +z +sqrt(1-2*z-3*z^2)).
T(n,k) = T(n-1,k-1)+ Sum_{j>=1} T(n-1,k+j) with T(0,0)=1. - Philippe Deléham, Jan 23 2010
T(n,k) = (k/n)*Sum_{j=k..n} (-1)^(n-j)*C(n,j)*C(2*j-k-1,j-1), n>0. - Vladimir Kruchinin, Feb 05 2011
From Emanuele Munarini, Jul 14 2024: (Start)
T(n,k) = Sum_{i=0..floor((n-k)/2)} binomial(n,i)*binomial(n-k-i-1,i-1)*(k+1)/(n-i+1).
T(n,k) = Sum_{i=0..n-k} (-1)^i*binomial(n,i)*binomial(2*n-k-2*i,n-i)*(k+1)/(n-i+1).
T(n,k) = (k+1)/(n+1)*Sum_{i=0..n-k} binomial(2*n-k-i,n)*trinomial(n+1,i)*(-1)^i, where trinomial(n,k) are the trinomial coefficients (A027907).
T(n,k) = Sum_{i=0..n-k} (-1)^i*binomial(2*n-k-i,n)*trinomial(n,i)*(i+k+1)/(n+1).
Recurrence: T(n+1,k+2) = T(n+1,k+1) - T(n,k+2) + T(n,k+1) - T(n,k). (End)

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

Original entry on oeis.org

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

Previous Showing 61-70 of 108 results. Next

cp's OEIS Frontend

A055249 Triangle of partial row sums (prs) of triangle A055248 (prs of Pascal's triangle A007318).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A093564 (7,1) Pascal triangle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Formula

A189391 The minimum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 0 to n, and each number in a higher row is the sum of the two numbers directly below it.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A054336 A convolution triangle of numbers based on A001405 (central binomial coefficients).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A054341 Row sums of triangle A054336 (central binomial convolutions).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A096956 Pascal (1,6) triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A008313 Triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

Views