A052509 -id:A052509 - cp's OEIS frontend

A194005 Triangle of the coefficients of an (n+1)-th order differential equation associated with A103631.

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

Offset: 0

Johannes W. Meijer and A. Hirschberg (a.hirschberg(AT)tue.nl), Aug 11 2011

This triangle is a companion to Parks' triangle A103631.
The coefficients of triangle A103631(n,k) appear in appendix 2 of Park’s remarkable article “A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov” if we assume that the b(n) coefficients are all equal to 1, see the second Maple program.
The a(n,k) coefficients of the triangle given above are related to the coefficients of a linear (n+1)-th order differential equation for the case b(n)=1, see the examples.
a(n,k) is also the number of symmetric binary strings of odd length n with Hamming weight k>0 and no consecutive 1's. - Christian Barrientos and Sarah Minion, Feb 27 2018

			For the 5th-order linear differential equation the coefficients a(k) are: a(0) = 1, a(1) = a(4,0) = 1, a(2) = a(4,1) = 4, a(3) = a(4,2) = 3, a(4) = a(4,3) = 3 and a(5) = a(4,4) = 1.
The corresponding Hurwitz matrices A(k) are, see Parks: A(5) = Matrix([[a(1),a(0),0,0,0], [a(3),a(2),a(1),a(0),0], [a(5),a(4),a(3),a(2),a(1)], [0,0,a(5),a(4),a(3)], [0,0,0,0,a(5)]]), A(4) = Matrix([[a(1),a(0),0,0], [a(3),a(2),a(1),a(0)], [a(5),a(4),a(3),a(2)], [0,0,a(5),a(4)]]), A(3) = Matrix([[a(1),a(0),0], [a(3),a(2),a(1)], [a(5),a(4),a(3)]]), A(2) = Matrix([[a(1),a(0)], [a(3),a(2)]]) and A(1) = Matrix([[a(1)]]).
The values of b(k) are, see Parks: b(1) = d(1), b(2) = d(2)/d(1), b(3) = d(3)/(d(1)*d(2)), b(4) = d(1)*d(4)/(d(2)*d(3)) and b(5) = d(2)*d(5)/(d(3)*d(4)).
These a(k) values lead to d(k) = 1 and subsequently to b(k) = 1 and this confirms our initial assumption, see the comments.
'
Triangle starts:
  [0] 1;
  [1] 1, 1;
  [2] 1, 2, 1;
  [3] 1, 3, 2,  1;
  [4] 1, 4, 3,  3,  1;
  [5] 1, 5, 4,  6,  3,  1;
  [6] 1, 6, 5, 10,  6,  4,  1;
  [7] 1, 7, 6, 15, 10, 10,  4,  1;
  [8] 1, 8, 7, 21, 15, 20, 10,  5, 1;
  [9] 1, 9, 8, 28, 21, 35, 20, 15, 5, 1;

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with corrections.
P.C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov , Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) p. 694-702.
Chris Zheng, Jeffrey Zheng, Triangular Numbers and Their Inherent Properties, Variant Construction from Theoretical Foundation to Applications, Springer, Singapore, 51-65.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A065941 and A103631.
Triangle sums (see A180662): A000071 (row sums; alt row sums), A075427 (Kn22), A000079 (Kn3), A109222(n+1)-1 (Kn4), A000045 (Fi1), A034943 (Ca3), A001519 (Gi3), A000930 (Ze3)
Interesting diagonals: T(n,n-4) = A189976(n+5) and T(n,n-5) = A189980(n+6)
Cf. A052509.

a194005 n k = a194005_tabl !! n !! k
a194005_row n = a194005_tabl !! n
a194005_tabl = [1] : [1,1] : f [1] [1,1] where
   f row' row = rs : f row rs where
     rs = zipWith (+) ([0,1] ++ row') (row ++ [0])
-- Reinhard Zumkeller, Nov 22 2012

A194005 := proc(n, k): binomial(floor((2*n+1-k)/2), n-k) end:
for n from 0 to 11 do seq(A194005(n, k), k=0..n) od;
seq(seq(A194005(n,k), k=0..n), n=0..11);
nmax:=11: for n from 0 to nmax+1 do b(n):=1 od:
A103631 := proc(n,k) option remember: local j: if k=0 and n=0 then b(1)
elif k=0 and n>=1 then 0 elif k=1 then b(n+1) elif k=2 then b(1)*b(n+1)
elif k>=3 then expand(b(n+1)*add(procname(j,k-2), j=k-2..n-2)) fi: end:
for n from 0 to nmax do for k from 0 to n do
A194005(n,k):= add(A103631(n1,k), n1=k..n) od: od:
seq(seq(A194005(n,k),k=0..n), n=0..nmax);

Flatten[Table[Binomial[Floor[(2n+1-k)/2],n-k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Apr 15 2012 *)

T(n,k) = binomial(floor((2*n+1-k)/2), n-k).
T(n,k) = sum(A103631(n1,k), n1=k..n), 0<=k<=n and n>=0.
T(n,k) = sum(binomial(floor((2*n1-k-1)/2), n1-k), n1=k..n).
T(n,0) = T(n,n) = 1, T(n,k) = T(n-2,k-2) + T(n-1,k), 0 < k < n. - Reinhard Zumkeller, Nov 23 2012

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

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

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

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

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

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

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

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

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

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

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

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

A054126 Odd-index Fibonacci row-sum array: T(n,k)=U(2n+1,n+1+k), 0<=k<=n, n >= 0, U the array in A054125.

Original entry on oeis.org

2, 3, 2, 6, 5, 2, 12, 13, 7, 2, 24, 30, 24, 9, 2, 48, 65, 68, 39, 11, 2, 96, 136, 171, 134, 58, 13, 2, 192, 279, 398, 394, 236, 81, 15, 2, 384, 566, 880, 1040, 802, 382, 108, 17, 2, 768, 1141, 1880, 2542, 2396, 1479, 580, 139, 19, 2, 1536

Offset: 0

Clark Kimberling

			Rows:
   2;
   3,  2;
   6,  5, 2;
  12, 13, 7, 2;
  ...

T(n,k) = if(k==n, 2, 2^(n-1-k) + sum(m=0, n-k, binomial(n+k, m))) \\ Jianing Song, May 30 2022

From Jianing Song, May 30 2022: (Start)
T(n,k) = 2 if k = n, otherwise A052509(2n,n+1+k) + A052509(2n,n-k) = 2^(n-1-k) + Sum_{m=0..n-k} binomial(n+k,m) = 2^(n-1-k) + 2^(n+k) - Sum_{m=0..2*k-1} binomial(n+k,m).
T(n,k) = [x^n*y^(n-k)] (1-x*y) * ((1+y-x*y^2)/((1-x*y^2)*((1-x*y)^2-x)) + (1+y-x*y)/((1-x)*((1-x*y)^2-x*y^2))). (End)

A053221 Row sums of triangle A053218.

Original entry on oeis.org

1, 5, 16, 43, 106, 249, 568, 1271, 2806, 6133, 13300, 28659, 61426, 131057, 278512, 589807, 1245166, 2621421, 5505004, 11534315, 24117226, 50331625, 104857576, 218103783, 452984806, 939524069, 1946157028, 4026531811, 8321499106

Offset: 1

Asher Auel, Jan 01 2000

Considered as a vector, the sequence = A074909 * [1, 2, 3, ...], where A074909 is the beheaded Pascal's triangle as a matrix. - Gary W. Adamson, Mar 06 2012
a(n) is the sum of the upper left n X n subarray of A052509 (viewed as an infinite square array). For example (1+1+1) + (1+2+2) + (1+3+4) = 16. - J. M. Bergot, Nov 06 2012
Number of ternary strings of length n that contain at least one 2 and at most one 0. For example, a(3) = 16 since the strings are the 6 permutations of 201, the 3 permutations of 211, the 3 permutations of 220, the 3 permutations of 221, and 222. - Enrique Navarrete, Jul 25 2021

			a(4) = 4 + 7 + 12 + 20 = 43.

Michael De Vlieger, Table of n, a(n) for n = 1..3311
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A053218, A053219, A053220.

Cf. A006127, A074909.

[(n+2)*2^(n-1)-n-1: n in [1..50]]; // G. C. Greubel, Sep 03 2018

A053221 := proc(n) (n+2)*2^(n-1)-n-1 ; end proc: # R. J. Mathar, Sep 02 2011

Table[(n + 2)*2^(n - 1) - n - 1, {n, 29}] (* or *)
Rest@ CoefficientList[Series[-x (-1 + x + x^2)/((2 x - 1)^2*(x - 1)^2), {x, 0, 29}], x] (* Michael De Vlieger, Sep 22 2017 *)
LinearRecurrence[{6,-13,12,-4},{1,5,16,43},30] (* Harvey P. Dale, Jun 28 2021 *)

vector(50,n, (n+2)*2^(n-1)-n-1) \\ G. C. Greubel, Sep 03 2018

a(n) = (n+2)*2^(n-1)-n-1. - Vladeta Jovovic, Feb 28 2003
G.f.: -x*(-1+x+x^2) / ( (2*x-1)^2*(x-1)^2 ). - R. J. Mathar, Sep 02 2011
a(n) = (1/2) * Sum_{k=1..n} Sum_{i=1..n} C(k,i) + C(n,k). - Wesley Ivan Hurt, Sep 22 2017
E.g.f.: exp(x)*(exp(x)-1)*(1+x). - Enrique Navarrete, Jul 25 2021
a(n+1) = 2*a(n) + A006127(n). - Ya-Ping Lu, Jan 01 2024

A054124 Left Fibonacci row-sum array, n >= 0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 4, 1, 1, 1, 2, 4, 7, 5, 1, 1, 1, 2, 4, 8, 11, 6, 1, 1, 1, 2, 4, 8, 15, 16, 7, 1, 1, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 1, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2

Offset: 0

Clark Kimberling

Reflection of array in A054123 about vertical central line.
Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:
  (1)  if x is in g(n-1), then x+1 is in g(n); and
  (2)  if x is in g(n-1) and x < 2, then x/2 is in g(n).
Then g(1) = {1/1}, g(2) = {1/2,2/1}, g(3) = {1/4,3/2,3/1}, etc. The denominators in g(n) are 2^0, 2^1, ..., 2^(n-1), and T(n,k) is the number of occurrences of 2^k, for k = 0..n-1. - Clark Kimberling, Nov 09 2015
Variant of A004070 with an additional column of 1's on the left. - Jianing Song, May 30 2022

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

Row sums: A000045. Central numbers: 1, 1, 2, 4, 8, ... (A000079).

First n numbers of n-th column for n >= 1 form the array in A008949.

a054124 n k = a054124_tabl !! n !! k
a054124_row n = a054124_tabl !! n
a054124_tabl = map reverse a054123_tabl
-- Reinhard Zumkeller, May 26 2015

t[, 0|1] = t[n, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-2, k-1]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 25 2013 *)

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

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

G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = (1-x^2*y) / ((1-x)*(1-x*y-x^2*y)). - Jianing Song, May 30 2022

A052511 a(n) = prime(n) - 1 - A006218(n).

Original entry on oeis.org

0, -1, -1, -2, 0, -2, 0, -2, -1, 1, 1, 1, 3, 1, 1, 2, 6, 2, 6, 4, 2, 4, 6, 4, 9, 9, 7, 5, 5, 1, 13, 11, 13, 11, 17, 10, 14, 16, 16, 14, 18, 12, 20, 16, 14, 12, 22, 24, 25, 21, 21, 21, 21, 23, 25, 23, 25, 23, 27, 19, 19, 25, 33, 30, 28, 24, 36, 36, 42, 36, 38, 32, 38, 40, 40, 38, 40, 40, 42, 40, 45, 43

Offset: 1

David Applegate and N. J. A. Sloane, Oct 14 2008

sign

a(n) > 0 for n >= 10.

The old entry with this sequence number was a duplicate of A052509.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..1000

See A088526 for another version.

from math import isqrt
from sympy import prime
def A052511(n): return prime(n)-1+(s:=isqrt(n))**2-(sum(n//k for k in range(1,s+1))<<1)  # Chai Wah Wu, Mar 26 2025

A164925 Array, binomial(j-i,j), read by rising antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 0, 1, 1, -2, 0, 0, 1, 1, -3, 1, 0, 0, 1, 1, -4, 3, 0, 0, 0, 1, 1, -5, 6, -1, 0, 0, 0, 1, 1, -6, 10, -4, 0, 0, 0, 0, 1, 1, -7, 15, -10, 1, 0, 0, 0, 0, 1, 1, -8, 21, -20, 5, 0, 0, 0, 0, 0, 1, 1, -9, 28, -35, 15, -1, 0, 0, 0, 0, 0, 1, 1, -10, 36, -56, 35, -6, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Mark Dols, Aug 31 2009

Inverse of A052509, or A004070???

			Array, A(n, k), begins as:
  1,  1,  1,   1,  1,   1,  1,  1,  1, ...
  1,  0,  0,   0,  0,   0,  0,  0,  0, ...
  1, -1,  0,   0,  0,   0,  0,  0,  0, ...
  1, -2,  1,   0,  0,   0,  0,  0,  0, ...
  1, -3,  3,  -1,  0,   0,  0,  0,  0, ...
  1, -4,  6,  -4,  1,   0,  0,  0,  0, ...
  1, -5, 10, -10,  5,  -1,  0,  0,  0, ...
  1, -6, 15, -20, 15,  -6,  1,  0,  0, ...
  1, -7, 21, -35, 35, -21,  7, -1,  0, ...
Antidiagonal triangle, T(n, k), begins as:
  1;
  1,  1;
  1,  0,  1;
  1, -1,  0,  1;
  1, -2,  0,  0,  1;
  1, -3,  1,  0,  0,  1;
  1, -4,  3,  0,  0,  0,  1;
  1, -5,  6, -1,  0,  0,  0,  1;
  1, -6, 10, -4,  0,  0,  0,  0,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A004070, A008346, A010892, A052509, A052992.

Cf. A109466, A130595, A164899, A164915, A164965.

A164925:= func< n,k | k eq 0 or k eq n select 1 else Binomial(2*k-n,k) >;
[A164925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2023

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[2*k-n, k]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 10 2023 *)

{A(i, j) = if( i<0, 0, if(i==0 || j==0, 1, binomial(j-i, j)))}; /* Michael Somos, Jan 25 2012 */

def A164925(n,k): return 1 if (k==0 or k==n) else binomial(2*k-n, k)
flatten([[A164925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 10 2023

Sum_{k=0..n} T(n, k) = A164965(n). - Mark Dols, Sep 02 2009
From G. C. Greubel, Feb 10 2023: (Start)
A(n, k) = binomial(k-n, k), with A(0, k) = A(n, 0) = 1 (array).
T(n, k) = binomial(2*k-n, k), with T(n, 0) = T(n, n) = 1 (antidiagonal triangle).
Sum_{k=0..n} (-1)^k*T(n, k) = A008346(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^n*A052992(n). (End)

Edited by Michael Somos, Jan 26 2012

Offset changed by G. C. Greubel, Feb 10 2023

A193605 Triangle: (row n) = partial sums of partial sums of row n of Pascal's triangle.

Original entry on oeis.org

1, 1, 3, 1, 4, 8, 1, 5, 12, 20, 1, 6, 17, 32, 48, 1, 7, 23, 49, 80, 112, 1, 8, 30, 72, 129, 192, 256, 1, 9, 38, 102, 201, 321, 448, 576, 1, 10, 47, 140, 303, 522, 769, 1024, 1280, 1, 11, 57, 187, 443, 825, 1291, 1793, 2304, 2816, 1, 12, 68, 244, 630, 1268, 2116, 3084, 4097, 5120, 6144

Offset: 0

Clark Kimberling, Jul 31 2011

The n-th row is contains the partial sums of the n-th row of the array interpretation of A052509. - R. J. Mathar, Apr 22 2013

			First 5 rows of A193605:
1
1....3
1....4....8
1....5....12....20
1....6....17....32....48

Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.

Cf. A193606.

A052509 := proc(n,k)
    if k = 0 then
        1;
    else
        procname(n,k-1)+binomial(n,k) ;
    end if;
end proc:
A193605 := proc(n,k)
    if k = 0 then
        1;
    else
        procname(n,k-1)+A052509(n,k) ;
    end if;
end proc: # R. J. Mathar, Apr 22 2013
# Alternative after Vladimir Kruchinin:
gf := ((x*y-1)/(1-2*x*y))^2/(1-x*y-x): ser := series(gf, x, 12):
p := n -> coeff(ser,x,n): row := n -> seq(coeff(p(n),y,k), k=0..n):
seq(row(n), n=0..10); # Peter Luschny, Aug 19 2019

u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}]
p[n_, k_] := Sum[u[n, h], {h, 0, k}]
Table[p[n, k], {n, 0, 12}, {k, 0, n}]
Flatten[%]   (* A193605 as a sequence *)
TableForm[Table[p[n, k], {n, 0, 12}, {k, 0, n}]]  (* A193605 as a triangle *)

T(n,k):=sum(((i+3)*2^(i-2))*binomial(n-i,k-i),i,1,min(n,k))+binomial(n,k);
/* Vladimir Kruchinin, Aug 20 2019 */

Writing the general term as T(n,k), for 0<=k<=n:
T(n,n)=A001792, T(n,n-1)=A001787, T(n,n-2)=A000337, T(n,n-3)=A045618.
T(n-1,k-1) + T(n-1,k) = T(n,k). - David A. Corneth, Oct 18 2016
G.f.: -(1-x*y)^2/(4*x^3*y^3+(4*x^3-8*x^2)*y^2+(5*x-4*x^2)*y+x-1). - Vladimir Kruchinin, Aug 19 2019
T(n,k) = C(n,k)+Sum_{i=1..n} (i+3)*2^(i-2)*C(n-i,k-i), - Vladimir Kruchinin, Aug 20 2019

More terms from David A. Corneth, Oct 18 2016

A054125 Sum of the arrays in A054123 and A054124.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 4, 4, 4, 2, 2, 5, 6, 6, 5, 2, 2, 6, 9, 8, 9, 6, 2, 2, 7, 13, 12, 12, 13, 7, 2, 2, 8, 18, 19, 16, 19, 18, 8, 2, 2, 9, 24, 30, 24, 24, 30, 24, 9, 2, 2, 10, 31, 46, 39, 32, 39, 46, 31, 10, 2, 2, 11, 39, 68, 65, 48, 48, 65

Offset: 0

Clark Kimberling

Row sums are twice Fibonacci numbers, A006355(n+2).

			Rows:
  2;
  2,2;
  2,2,2;
  2,3,3,2;
  ...

A052509(n,k) = sum(m=0, k, binomial(n-k, m));
T(n,k) = if(k==0 || k==n, 2, A052509(n-1,k) + A052509(n-1,n-k)) \\ Jianing Song, May 30 2022

From Jianing Song, May 30 2022: (Start)
T(n,k) = 2 if k = 0 or k = n, A052509(n-1,k) + A052509(n-1,n-k) otherwise.
G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = (1-x^2*y) * (1/((1-x*y)*(1-x-x^2*y)) + 1/((1-x)*(1-x*y-x^2*y))). (End)

A131250 A007318 * A004070.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 8, 11, 6, 1, 16, 26, 22, 8, 1, 32, 57, 64, 37, 10, 1, 64, 120, 163, 130, 56, 12, 1, 128, 247, 382, 386, 232, 79, 14, 1, 256, 502, 848, 1024, 794, 378, 106, 16, 1, 512, 1013, 1816, 2510, 2380, 1471, 576, 137, 18, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A061667: (1, 3, 9, 26, 73, 201, ...).
Companion triangle = A131249 = A007318 * A052509, where A052509 is the reversal of A004070.
Reversal of A097750. - Philippe Deléham, Jan 11 2014
Riordan array (1/(1-2x), x/(1-x)^2). - Philippe Deléham, Jan 11 2014
Diagonal sums are A045623. - Philippe Deléham, Jan 11 2014

			First few rows of the triangle:
   1;
   2,  1;
   4,  4,  1;
   8, 11,  6,  1;
  16, 26, 22,  8,  1;
  32, 57, 64, 37, 10,  1;
  ...

Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2 (see Table 2).

Cf. A004070, A061667, A131249, A007318.

Binomial transform of A004070.

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014

More terms from Philippe Deléham, Jan 11 2014

cp's OEIS Frontend

A194005 Triangle of the coefficients of an (n+1)-th order differential equation associated with A103631.

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

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A054126 Odd-index Fibonacci row-sum array: T(n,k)=U(2n+1,n+1+k), 0<=k<=n, n >= 0, U the array in A054125.

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A053221 Row sums of triangle A053218.

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A054124 Left Fibonacci row-sum array, n >= 0, 0<=k<=n.

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A052511 a(n) = prime(n) - 1 - A006218(n).

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A164925 Array, binomial(j-i,j), read by rising antidiagonals.

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