A055248 -id:A055248 - cp's OEIS frontend

1, 5, 1, 19, 6, 1, 63, 25, 7, 1, 192, 88, 32, 8, 1, 552, 280, 120, 40, 9, 1, 1520, 832, 400, 160, 49, 10, 1, 4048, 2352, 1232, 560, 209, 59, 11, 1, 10496, 6400, 3584, 1792, 769, 268, 70, 12, 1, 26624, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 66304, 43520

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)^3)/(1-2*z)^4)/(1-x*z/(1-z)).
This is the fourth 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 as A049612(n+1), A055585, A001794, A001789(n+3), A027608, A055586 for m=0..5.

			[0] 1
[1] 5, 1
[2] 19, 6, 1
[3] 63, 25, 7, 1
[4] 192, 88, 32, 8, 1
[5] 552, 280, 120, 40, 9, 1
[6] 1520, 832, 400, 160, 49, 10, 1
[7] 4048, 2352, 1232, 560, 209, 59, 11, 1
Fourth row polynomial (n=3): p(3, x)= 63 + 25*x + 7*x^2 + x^3.

Cf. A007318, A055248, A055249, A055252. Row sums: A049600(n+1, 4).

T := (n, k) -> binomial(n, k)*hypergeom([4, k - n], [k + 1], -1):
for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # Peter Luschny, Sep 23 2024

a(n, m)=sum(A055252(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j, m), j=m..n-1)+ A055252(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: (((1-x)^3)/(1-2*x)^4)*(x/(1-x))^m, m >= 0.

T(n, k) = binomial(n, k)*hypergeom([4, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

A104709 Triangle read by rows: T(n,k) = Sum_{j=0..n} 2^(n-j)binomial(j,k) for n >= 0 and 0 <= k <= n; also, Riordan array (1/((1-x)(1-2*x)), x/(1-x)).

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 15, 11, 5, 1, 31, 26, 16, 6, 1, 63, 57, 42, 22, 7, 1, 127, 120, 99, 64, 29, 8, 1, 255, 247, 219, 163, 93, 37, 9, 1, 511, 502, 466, 382, 256, 130, 46, 10, 1, 1023, 1013, 968, 848, 638, 386, 176, 56, 11, 1, 2047, 2036, 1981, 1816, 1486, 1024, 562, 232, 67

Offset: 0

Gary W. Adamson, Mar 19 2005

This array (A104709) is the mirror of the fission, A054143, of the polynomial sequence ((x+1)^n: n >= 0) by the polynomial sequence (q(n,x): n >= 0) given by q(n,x) = x^n + x^(n-1) + ... + x + 1.  See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011
The elements of the matrix inverse appear to be T^(-1)(n,k) = (-1)^(n+k)*A110813(n,k) assuming the same offset in both triangles. - R. J. Mathar, Mar 15 2013
From Paul Curtz, Jun 12 2019: (Start)
Numerators of the triangle [Curtz, page 15, triangle (E)]:
     1/2;
     3/4,  1/4;
     7/8,  4/8,    1/8;
   15/16, 11/16,  5/16,  1/16;
   31/32, 26/31, 16/32,  6/32, 1/32;
   63/64, 57/64, 42/64, 22/64, 7/64, 1/64;
   ...
Denominators - Numerators: Triangle A054143.
  1;
  1, 3;
  1, 4, 7;
  1, 5, 11, 15;
  ...
(E) is a transform which accelerates the convergence of series.
For log(2) = 1 - 1/2 + 1/3 - 1/4 ... = 0.6931..., we have
  1*(1/2) = 1/2,
  1*(3/4) - (1/2)*(1/4) = 5/8,
  1*(7/8) - (1/2)*(4/8) + (1/3)*(1/8) = 2/3,
  1*(15/16) - (1/2)*(11/16) + (1/3)*(5/16) - (1/4)*1/16 = 131/192,
  ...
This is A068566/A068565. (End)

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
   1;
   3,  1;
   7,  4,  1;
  15, 11,  5,  1;
  31, 26, 16,  6,  1;
  63, 57, 42, 22,  7,  1;
  ...

Paul Curtz, Accélération de la convergence de certaines séries alternées à l'aide des fonctions de sommation, Thèse de 3ème Cycle d'Analyse Numérique, Faculté des Sciences de l'Université de Paris, 4 mai 1965.
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52(3) (2014), 195-202.

Cf. A000124, A000295, A001787 (row sums), A007318, A054143, A055248, A068565, A068566, A104709, A140513.

A104709_row := proc(n) add(add(binomial(n,n-i)*x^(n-k-1),i=0..k),k=0..n-1);
coeffs(sort(%)) end; seq(print(A104709_row(n)),n=1..6); # Peter Luschny, Sep 29 2011

z = 10;
p[n_, x_] := (x + 1)^n;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A054143 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]] (* A104709 *)
(* Clark Kimberling, Aug 07 2011 *)

Begin with A055248 as a triangle, delete leftmost column.
The Riordan array factors as (1/(1-2*x), x)*(1/(1-x), x/(1-x)) - the sequence array for 2^n times Pascal's triangle. - Paul Barry, Aug 05 2005
T(n,k) = Sum_{j=0..n-k} C(n-j, k)*2^j. - Paul Barry, Jan 12 2006
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 2*T(n-2,k-1), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 30 2013
Working with an offset of 0, we have exp(x) * (e.g.f. for row n) = (e.g.f. for diagonal n). For example, for n = 3 we have exp(x)*(15 + 11*x + 5*x^2/2! + x^3/3!) = 15 + 26*x + 42*x^2/2! + 64*x^3/3! + 93*x^4/4! + .... The same property holds more generally for Riordan arrays of the form (f(x), x/(1 - x)). - Peter Bala, Dec 21 2014
From Petros Hadjicostas, Jun 05 2020: (Start)
Bivariate o.g.f.: A(x,y) = Sum_{n,k >= 0} T(n,k)*x^n*y^k = 1/(1 - 3*x - x*y + 2*x^2 + 2*x^2*y) = 1/((1 - 2*x)*(1 - x*(y+1))).
The o.g.f. of the n-th row is (2^(n+1) - (1 + y)^(n+1))/(1 - y).
Let B(x,y) be the bivariate o.g.f. of triangular array A054143. Because A054143 is the mirror image of the current array, we have A(x,y) = B(x*y, 1/y) and B(x,y) = A(x*y, 1/y). This makes it easy to identify lower diagonals of the array.
For example, if we want to identify the second lower diagonal of the array (i.e., 7, 11, 16, 22, ...), we take the 2nd derivative of B(x,y) with respect to y, set y = 0, and divide by 2!. (Note that columns in A054143 start at k = 0.) We get the g.f. x^2*(7 - 10*x + 4*x^2)/(1 - x)^3.
It is then easy to derive that T(n,n-2) = A000124(n+1) = (n+1)*(n+2)/2 + 1 for n >= 2 (by ignoring the first three terms of A000124). Of course, in the current case, it is much easier to use the formula for T(n,k) to find T(n,n-2). (End)
T(n,0) = 2^(n+1) - 1 for n >= 0; T(n,k) = T(n-1,k) + T(n-1,k-1) for 1 <= k <= n. - Peter Bala, Jan 30 2023
T(n,1) = 2^(n+1) - n - 2 = A000295(n+1) for n >= 1. - Bernard Schott, Feb 22 2023

Name edited and offset changed by Petros Hadjicostas, Jun 04 2020

A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 3, 1, 0, 1, 2, 4, 4, 1, 1, 1, 2, 4, 7, 5, 1, 0, 1, 2, 4, 8, 11, 6, 1, 1, 1, 2, 4, 8, 15, 16, 7, 1, 0, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 0, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 0

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(0,2n)=T(1,n) by T(0,2n)=T(m,n) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058394, A058395, A057884, (and effectively A007318).

			Rows are (1,0,1,0,1,0,1,...), (1,1,1,1,1,1,...), (1,2,2,2,2,2,...), (1,3,4,4,4,...) etc.

Rows are A000035 (A000012 with zeros), A000012, A040000 etc. Columns are A000012, A001477, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863 etc. Diagonals include A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, etc. The triangles A008949, A054143 and A055248 also appear in the half of the array which is not powers of 2.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(1, 1)=1, T(0, 2n)=T(1, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2).

A061930 Square array read by antidiagonals of T(n,k)=T(n-1,[k/2])+T(n-1,[k/3]) with T(0,0)=1.

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 1, 4, 8, 0, 0, 4, 8, 16, 0, 0, 4, 8, 16, 32, 0, 0, 3, 8, 16, 32, 64, 0, 0, 3, 8, 16, 32, 64, 128, 0, 0, 1, 8, 16, 32, 64, 128, 256, 0, 0, 1, 8, 16, 32, 64, 128, 256, 512, 0, 0, 1, 8, 16, 32, 64, 128, 256, 512, 1024, 0, 0, 0, 7, 16, 32, 64, 128, 256, 512, 1024

Offset: 0

Henry Bottomley, May 22 2001

			T(9, 7) = T(8, [7/2])+T(8, [7/3]) = T(8, 3)+T(8, 2) = 256+256 = 512. Rows start (1, 0, 0, 0, 0, ...), (2, 2, 1, 0, 0, ...), (4, 4, 4, 4, 3, ...) etc.

Row sums are 5^n, i.e. A000351. Each row starts with 2^n copies of 2^n, i.e. A000079 and then continues with A036561 copies of other terms in the rows of A055248. Cf. A061929.

A215079 Triangle T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k).

Original entry on oeis.org

1, 0, 1, 0, 3, 4, 0, 7, 32, 27, 0, 15, 176, 405, 256, 0, 31, 832, 3888, 6144, 3125, 0, 63, 3648, 30618, 90112, 109375, 46656, 0, 127, 15360, 216513, 1048576, 2265625, 2239488, 823543, 0, 255, 63232, 1436859, 10682368, 36328125, 62145792, 51883209, 16777216, 0, 511, 257024, 9172278, 100139008, 500000000, 1310100480, 1856265922, 1342177280, 387420489, 0, 1023, 1037312, 57159432, 889192448, 6230468750, 23339943936, 49715643824, 60129542144, 38354628411, 10000000000

Offset: 0

Olivier Gérard, Aug 02 2012

Initial term T(0,0) may be computed as 0, depending on formula and convention.

			      1
      0       1
      0       3       4
      0       7      32      27
      0      15     176     405     256
      0      31     832    3888    6144    3125
      0      63    3648   30618   90112  109375   46656
      0     127   15360  216513 1048576 2265625 2239488  823543

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Row sums sequence is A215077.
Product of A055248 and A089072 (with an initial 0 in each row).
Cf. A000225 (column k=1), A000312 (diagonal).

A215079 := proc(n,k)
    k^n*add( binomial(n,n-k-j),j=0..n-k) ;
end proc: # R. J. Mathar, Feb 08 2021

Flatten[Table[Table[Sum[k^n*Binomial[n, n - k - j], {j, 0, n - k}],  {k, 0, n}], {n, 0, 10}], 1]

T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k) = k^n * A055248(n,k-1).
T(n,k) = k^n * binomial(n,n-k) * 2F1(1, k-n; k+1)(-1)
T(n,1) = A000225(n). - R. J. Mathar, Feb 08 2021

A103316 Riordan array (1/(1+2x), x/(1+x)).

Original entry on oeis.org

1, -2, 1, 4, -3, 1, -8, 7, -4, 1, 16, -15, 11, -5, 1, -32, 31, -26, 16, -6, 1, 64, -63, 57, -42, 22, -7, 1, -128, 127, -120, 99, -64, 29, -8, 1, 256, -255, 247, -219, 163, -93, 37, -9, 1, -512, 511, -502, 466, -382, 256, -130, 46, -10, 1, 1024, -1023, 1013, -968, 848, -638, 386, -176, 56, -11, 1, -2048, 2047, -2036, 1981

Offset: 0

Paul Barry, Jan 30 2005

Inverse array of A029653. Signed version of A055248. Row sums are (-1)^n*A011782(n), with g.f. (1+x)/(1+2x). Diagonal sums are (-1)^n*A027934(n), with g.f. (1+x)/((1+2x)(1+x-x^2)).

			Rows start {1}, {-2,1}, {4,-3,1}, {-8,7,-4,1},...

Number triangle T(n, k)=(-1)^(n-k)*sum{j=0..n, binomial(n, k+j)}

T(n,k)=T(n-1,k-1)-3*T(n-1,k)+2*T(n-2,k-1)-2*T(n-2,k), T(0,0)=1, T(1,0)=-2, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 12 2014

A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 4, -1, 1, 11, -5, 1, 1, 26, -16, 6, -1, 1, 57, -42, 22, -7, 1, 1, 120, -99, 64, -29, 8, -1, 1, 247, -219, 163, -93, 37, -9, 1, 1, 502, -466, 382, -256, 130, -46, 10, -1, 1, 1013, -968, 848, -638, 386, -176, 56, -11, 1, 2036, -1981, 1816, -1486, 1024

Offset: 0

Philippe Deléham, Nov 30 2013

Row sums are A000079(n) = 2^n.
Diagonal sums are A024493(n+1) = A130781(n).
Sum_{k=0..n} T(n,k)*x^k = -A003063(n+2), A159964(n), A000012(n), A000079(n), A001045(n+2), A056450(n), (-1)^(n+1)*A232015(n+1) for x = -2, -1, 0, 1, 2, 3, 4 respectively.

			Triangle begins:
  1;
  1,    1;
  1,    4,   -1;
  1,   11,   -5,   1;
  1,   26,  -16,   6,   -1;
  1,   57,  -42,  22,   -7,   1;
  1,  120,  -99,  64,  -29,   8,   -1;
  1,  247, -219, 163,  -93,  37,   -9,  1;
  1,  502, -466, 382, -256, 130,  -46, 10,  -1;
  1, 1013, -968, 848, -638, 386, -176, 56, -11, 1;

Columns: A000012, A000295, -A002662, A002663, -A002664, A035038, -A035039, A035040, -A035041, A035042.
Cf. A000079, A055248, A024493, A130781, A000302.
Cf. also A003063, A159964, A000012, A001045, A056450, A232015.

G.f.: Sum_{n>=0, k=0..n} T(n,k)*y^k*x^n=(1+2*(y-1)*x)/((1-2*x)*(1+(y-1)*x)).
|T(2*n,n)| = 4^n = A000302(n).
T(n,k) = (-1)^(k-1) * (Sum_{i=0..n-k} (2^(i+1)-1) * binomial(n-i-1,k-1)) for 0 < k <= n and T(n,0) = 1 for n >= 0. - Werner Schulte, Mar 22 2019

A348451 Triangle read by rows: T(n,k) (1 <= k <= n) is the number of 3-extensions of an n-set over all choices of 3-partitions of the n-set.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 14, 13, 6, 1, 41, 40, 25, 9, 1, 122, 121, 90, 48, 12, 1, 365, 364, 301, 202, 78, 14, 1, 1094, 1093, 966, 747, 380, 106, 16, 1, 3281, 3280, 3025, 2559, 1571, 592, 141, 18, 1, 9842, 9841, 9330, 8362, 5864, 2755, 906, 180, 20, 1

Offset: 1

N. J. A. Sloane, Oct 26 2021

See Lindquist et al. 1981 for precise definition.

			Triangle begins:
1,
2,1,
5,4,1,
14,13,6,1,
41,40,25,9,1,
122,121,90,48,12,1,
365,364,301,202,78,14,1,
1094,1093,966,747,380,106,16,1,
3281,3280,3025,2559,1571,592,141,18,1,
9842,9841,9330,8362,5864,2755,906,180,20,1,
...

Norman Lindquist and Gerard Sierksma, Extensions of set partitions, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table III.

Cf. A055248.

Column 1 = A007051, column 3 = A000392.

Previous Showing 21-30 of 30 results.

cp's OEIS Frontend

A035041 a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,8).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A112626 Triangle read by rows: T(n,k) = Sum_{j=0..n} binomial(n, k+j)*2^(n-k-j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A055584 Triangle of partial row sums (prs) of triangle A055252.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A104709 Triangle read by rows: T(n,k) = Sum_{j=0..n} 2^(n-j)*binomial(j,k) for n >= 0 and 0 <= k <= n; also, Riordan array (1/((1-x)*(1-2*x)), x/(1-x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A061930 Square array read by antidiagonals of T(n,k)=T(n-1,[k/2])+T(n-1,[k/3]) with T(0,0)=1.

Views

Author

Keywords

Examples

Crossrefs

A215079 Triangle T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A103316 Riordan array (1/(1+2x), x/(1+x)).

Views

A104709 Triangle read by rows: T(n,k) = Sum_{j=0..n} 2^(n-j)binomial(j,k) for n >= 0 and 0 <= k <= n; also, Riordan array (1/((1-x)(1-2*x)), x/(1-x)).