A030662 -id:A030662 - cp's OEIS frontend

A260877 Square array read by ascending antidiagonals: number of m-shape Euler numbers.

1, 1, -1, 1, -1, 1, 1, -1, 1, -5, 1, -1, 5, -1, 21, 1, -1, 19, -61, 1, -105, 1, -1, 69, -1513, 1385, -1, 635, 1, -1, 251, -33661, 315523, -50521, 1, -4507, 1, -1, 923, -750751, 60376809, -136085041, 2702765, -1, 36457, 1, -1, 3431, -17116009, 11593285251

Offset: 1

Peter Luschny, Aug 09 2015

A set partition of m-shape is a partition of a set with cardinality m*n for some n >= 0 such that the sizes of the blocks are m times the parts of the integer partitions of n. It is ordered if the positions of the blocks are taken into account.
M-shape Euler numbers count the ordered m-shape set partitions which have even length minus the number of such partitions which have odd length.
If m=0 all possible sizes are zero. Thus m-shape Euler numbers count the ordered integer partitions of n into an even number of parts minus the number of ordered integer partitions of n into an odd number of parts (A260845).
If m=1 the set is {1,2,...,n} and the set of all possible sizes are the integer partitions of n. Thus the Euler numbers count the ordered set partitions which have even length minus the set partitions which have odd length (A033999).
If m=2 the set is {1,2,...,2n} and the 2-shape Euler numbers count the ordered set partitions with even blocks which have even length minus the number of partitions with even blocks which have odd length (A028296).

			[ n ] [0   1   2       3         4              5                 6]
[ m ] --------------------------------------------------------------
[ 0 ] [1, -1,  1,     -5,       21,          -105,              635] A260845
[ 1 ] [1, -1,  1,     -1,        1,            -1,                1] A033999
[ 2 ] [1, -1,  5,    -61,     1385,        -50521,          2702765] A028296
[ 3 ] [1, -1, 19,  -1513,   315523,    -136085041,     105261234643] A002115
[ 4 ] [1, -1, 69, -33661, 60376809, -288294050521, 3019098162602349] A211212
         A030662,A211213,  A181991,
For example the number of ordered set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] are 1, 168, 1680 respectively. Thus A(3,3) = -1 + 168 - 1680 = -1513.
Formatted as a triangle:
[1]
[1, -1]
[1, -1,  1]
[1, -1,  1,    -5]
[1, -1,  5,    -1,   21]
[1, -1, 19,   -61,    1, -105]
[1, -1, 69, -1513, 1385,   -1, 635]

Cf. A002115, A028296, A030662, A033999, A181991, A211212, A211213, A260845, A260833, A260875, A260876.

def A260877(m,n):
    shapes = ([x*m for x in p] for p in Partitions(n).list())
    return sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s). cardinality() for s in shapes)
for m in (0..5): print([A260877(m,n) for n in (0..7)])

A101475 Triangle T(n,k) read by rows: number of lattice paths from (0,0) to (0,2n) with steps (1,1) or (1,-1) that stay between the lines y=0 and y=k.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 15, 19, 20, 35, 50, 63, 69, 70, 126, 176, 217, 243, 251, 252, 462, 638, 770, 870, 913, 923, 924, 1716, 2354, 2794, 3159, 3355, 3419, 3431, 3432, 6435, 8789, 10307, 11610, 12430, 12766, 12855, 12869, 12870, 24310, 33099, 38489

Offset: 0

Ralf Stephan, Jan 21 2005

			Triangle begins
     1;
     1,    2;
     3,    5,     6;
    10,   15,    19,    20;
    35,   50,    63,    69,    70;
   126,  176,   217,   243,   251,   252;
   462,  638,   770,   870,   913,   923,   924;
  1716, 2354,  2794,  3159,  3355,  3419,  3431,  3432;
  6435, 8789, 10307, 11610, 12430, 12766, 12855, 12869, 12870;

W. Y. C. Cheng, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.

Left-hand columns include A001700 and A024718. Right-hand columns include A000984 and A030662. Row sums are in A101476.

T[n_, k_] := Sum[Binomial[2n, n-i(k+2)] - Binomial[2n, n+i(k+2)+k+1], {i, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2019 *)

T(n, k) = Sum_{i>=0} (binomial(2n, n-i*(k+2)) - binomial(2n, n+i*(k+2)+k+1)).

A128205 a(n) = 2^(n-1)*A047240(n).

Original entry on oeis.org

0, 1, 4, 24, 56, 128, 384, 832, 1792, 4608, 9728, 20480, 49152, 102400, 212992, 491520, 1015808, 2097152, 4718592, 9699328, 19922944, 44040192, 90177536, 184549376, 402653184, 822083584, 1677721600, 3623878656, 7381975040, 15032385536, 32212254720

Offset: 0

Paul Barry, Feb 19 2007

-a(n) is the Hankel transform of A030662(n) = binomial(2*n,n)-1.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,8,-16).

Cf. A047240, A030662.

a047240[n_] := 6 Floor[n/3] + Mod[n, 3]
a128205[n_] := Map[2^(#-1) a047240[#]&, Range[0, n]]
a128205[25] (* data *) (* Hartmut F. W. Hoft, Mar 13 2017 *)
LinearRecurrence[{2,0,8,-16},{0,1,4,24},40] (* Harvey P. Dale, Feb 13 2024 *)

concat(0, Vec(x*(1 + 2*x + 16*x^2) / ((1 - 2*x)^2*(1 + 2*x + 4*x^2)) + O(x^40))) \\ Colin Barker, Mar 13 2017

a(n) = 2^(n-1)*(cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3)/3 + 2n - 1);

O.g.f.: x(1+2x+16x^2)/((2x-1)^2*(4x^2+2x+1)). a(n) = 2a(n-1) + 8a(n-3) - 16a(n-4). - R. J. Mathar, Apr 28 2008

A172021 Start with the triangle A171661, reverse its rows, add missing powers of 2 at beginning of each row.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 4, 6, 6, 1, 2, 4, 8, 14, 20, 20, 1, 2, 4, 8, 16, 30, 50, 70, 70, 1, 2, 4, 8, 16, 32, 62, 112, 182, 252, 252, 1, 2, 4, 8, 16, 32, 64, 126, 238, 420, 672, 924, 924, 1, 2, 4, 8, 16, 32, 64, 128, 254, 492, 912, 1584, 2508, 3432, 3432

Offset: 1

Mark Dols, Jan 22 2010

Rows sum up to A030662.
Triangle is a (mirrored) interspaced binomial transform of 1^n (see example). - Mark Dols, Jan 24 2010
T(n,k) is the number of k permutations of n (indistinguishable) objects of type I and n (indistinguishable) objects of type II. - Geoffrey Critzer, Mar 15 2010
Equivalently T(n,k) is the number of words length k from an alphabet of 2 letters with at most n occurrences of each letter. - Giovanni Artico, Aug 24 2013
T(n,k) is also the number of ways k persons can be accommodated into 2 rooms with at most n persons per room. - Giovanni Artico, Aug 24 2013

			Triangle begins:
......1
....1,2,2
..1,2,4,6,6
1,2,4,8,14,20,20
From _Mark Dols_, Jan 24 2010: (Start)
Interspaced binomial transform of 1^n:
1...1...1...1...1...1...
..2...2...2...2...2...2.
2...4...4...4...4...4...
..6...8...8...8...8...8.
6.. 14..16..16..16..16..
..20..30..32..32..32..32
20..50..62..64..64..64..
(End)

Cf. A030662, A171661, A171698, A004070.

T(n,k):=POLY_COEFF(SUM(x^i/i!, i, 0, n)^2, x, k)·k!
TABLE(VECTOR(T(v, u), u, 0, 2·v), v, 0, 10)  # Giovanni Artico, Aug 30 2013

seq(PolynomialTools:-CoefficientList((convert(taylor(exp(x),x,n+1),polynom)^2),x)*~[seq(i!,i=0..2 n)],n=0..10) # Giovanni Artico, Aug 30 2013

Table[CoefficientList[Series[(Sum[x^i/i!, {i, 0, m}])^2, {x, 0, 2 m}], x]*Table[n!, {n, 0, 2 m}], {m, 0, 10}] // Grid (* Geoffrey Critzer, Mar 15 2010 *)

E.g.f. for row n is: ( 1 + x + x^2/2! + ... + x^n/n! )^2. - Geoffrey Critzer, Mar 15 2010

Definition rewritten by N. J. A. Sloane, Jan 23 2010

More terms from Mark Dols, Jan 24 2010

A363570 Expansion of (1 / sqrt(1 - 8x + 4x^2) - 1 / (1 - x)) / 3.

Original entry on oeis.org

0, 1, 7, 45, 295, 1981, 13545, 93829, 656311, 4625181, 32788657, 233567269, 1670457321, 11987269477, 86268665917, 622391877045, 4500029549911, 32598283556317, 236542093805025, 1719008077215205, 12509403045819505, 91143878730342021, 664816240262272237

Offset: 0

Ilya Gutkovskiy, Aug 17 2023

nonn

Cf. A030662, A047665, A069835, A098663, A363571.

nmax = 22; CoefficientList[Series[(1/Sqrt[1 - 8 x + 4 x^2] - 1/(1 - x))/3, {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Exp[x] (Exp[3 x] BesselI[0, 2 Sqrt[3] x] - 1)/3, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 3^(k - 1), {k, 1, n}], {n, 0, 22}]
Table[(2^n LegendreP[n, 2] - 1)/3, {n, 0, 22}]

E.g.f.: exp(x) * (exp(3*x) * BesselI(0,2*sqrt(3)*x) - 1) / 3.
a(n) = Sum_{k=1..n} binomial(n,k)^2 * 3^(k-1).
a(n) = (2^n * LegendreP(n,2) - 1) / 3.
a(n) = (A069835(n) - 1) / 3.

A363571 Expansion of (1 / sqrt(1 - 10x + 9x^2) - 1 / (1 - x)) / 4.

Original entry on oeis.org

0, 1, 8, 61, 480, 3881, 31976, 266981, 2251136, 19124881, 163452168, 1403748941, 12104113632, 104723793721, 908680775528, 7904234296181, 68905275700736, 601832985410081, 5265459181363976, 46137526574521181, 404821208100919520, 3556361565584509001

Offset: 0

Ilya Gutkovskiy, Aug 17 2023

nonn

Cf. A030662, A047665, A084771, A098665, A363570.

nmax = 21; CoefficientList[Series[(1/Sqrt[1 - 10 x + 9 x^2] - 1/(1 - x))/4, {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[Exp[x] (Exp[4 x] BesselI[0, 4 x] - 1)/4, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 4^(k - 1), {k, 1, n}], {n, 0, 21}]
Table[(3^n LegendreP[n, 5/3] - 1)/4, {n, 0, 21}]

E.g.f.: exp(x) * (exp(4*x) * BesselI(0,4*x) - 1) / 4.
a(n) = Sum_{k=1..n} binomial(n,k)^2 * 4^(k-1).
a(n) = (3^n * LegendreP(n,5/3) - 1) / 4.
a(n) = (A084771(n) - 1) / 4.

A369335 a(n) is the sum of the numbers inside a square of side length 2n+1 located in Pascal's triangle at and below row n.

Original entry on oeis.org

1, 10, 94, 816, 6872, 57038, 469238, 3836430, 31221874, 253198806, 2047761014, 16526046182, 133145419056, 1071276327274, 8610169465642, 69143029079432, 554860626424744, 4450160058135914, 35675446422203960, 285892025190834636, 2290356743575612582

Offset: 0

Nicolay Avilov, Jan 20 2024

nonn

We will assume that the numbers of Pascal's triangle are written in the cells of a square lattice. Then row n has width 2n+1 and the square of cells starts there.

			a(0) = 1.
a(1) = 1 + 1 + 2 + 3 + 3 = 10.
            -----------
           | 1   2   1 |
           |   3   3   |
a(2) = Sum | 4   6   4 | = 94.
           |   10  10  |
           | 15  20  15|
            -----------

Alois P. Heinz, Table of n, a(n) for n = 0..1106
Nicolay Avilov, Illustration for terms a(0) - a(4).

Cf. A007318, A030662.

a[n_]:=Sum[Sum[Binomial[i,k],{k,Floor[(i+1-n)/2],Floor[(i+1-n)/2]+n-Mod[i-n,2]}],{i,n,3n}]; Array[a,21,0] (* Stefano Spezia, Jan 21 2024 *)

Limit_{n->oo} a(n+1)/a(n) = 8.

a(6)-a(20) from Alois P. Heinz, Jan 20 2024

A375555 Triangle read by rows: T(n, k) = abs(A181937(k, n)), where A181937 are the André numbers, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 16, 9, 4, 1, 1, 1, 61, 19, 14, 5, 1, 1, 1, 272, 99, 34, 20, 6, 1, 1, 1, 1385, 477, 69, 55, 27, 7, 1, 1, 1, 7936, 1513, 496, 125, 83, 35, 8, 1, 1, 1, 50521, 11259, 2896, 251, 209, 119, 44, 9, 1

Offset: 0

Peter Luschny, Aug 19 2024

See A181937 for comments and references.

			Triangle starts:
  [0]  1;
  [1]  1, 1;
  [2]  1, 1,    1;
  [3]  1, 1,    2,    1;
  [4]  1, 1,    5,    3,   1;
  [5]  1, 1,   16,    9,   4,   1;
  [6]  1, 1,   61,   19,  14,   5,  1;
  [7]  1, 1,  272,   99,  34,  20,  6,  1;
  [8]  1, 1, 1385,  477,  69,  55, 27,  7, 1;
  [9]  1, 1, 7936, 1513, 496, 125, 83, 35, 8, 1;
.
Seen as an array:
  [0]  1, 1,      1,      1,      1,      1,      1,      1, ...
  [1]  1, 1,      2,      3,      4,      5,      6,      7, ...
  [2]  1, 1,      5,      9,     14,     20,     27,     35, ...
  [3]  1, 1,     16,     19,     34,     55,     83,    119, ...
  [4]  1, 1,     61,     99,     69,    125,    209,    329, ...
  [5]  1, 1,    272,    477,    496,    251,    461,    791, ...
  [6]  1, 1,   1385,   1513,   2896,   2300,    923,   1715, ...
  [7]  1, 1,   7936,  11259,  11056,  15775,  10284,   3431, ...

Cf. A181937, A375554 (row sums), A030662 (central terms, main diagonal of array), A010763 (central terms of the (1, 1)-based variant).

Andre := proc(n, k) option remember; local j;
  ifelse(k = 0, 1, ifelse(n = 0, 1,
  -add(binomial(k, j) * Andre(n, j), j = 0..k-1, n))) end:
T := (n, k) -> abs(Andre(k, n)): seq(seq(T(n, k), k = 0..n), n = 0..10);

Andre[n_, k_] := Andre[n, k] = If[k <= 0, 1, If[n == 0, 1, -Sum[Binomial[k, j] Andre[n, j], {j, 0, k-1, n}]]];
(* Seen as an array: *)
A[n_, k_] := Abs[Andre[k, n + k]];
Table[A[n, k], {n, 0, 9}, {k, 0, 7}] // MatrixForm

A085391 Square array of centered numbers, read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 4, 5, 1, 0, 1, 5, 10, 7, 1, 0, 1, 6, 15, 19, 9, 1, 0, 1, 7, 21, 35, 31, 11, 1, 0, 1, 8, 28, 56, 69, 46, 13, 1, 0, 1, 9, 36, 84, 126, 121, 64, 15, 1, 0, 1, 10, 45, 120, 210, 251, 195, 85, 17, 1, 0, 1, 11, 55, 165, 330, 462, 456, 295, 109, 19, 1, 0

Offset: 0

Paul Barry, Jul 02 2003

			Rows begin
0 0 0 0 0 0 ...
1 1 1 1 1 1 ...
1 3 5 7 9 11 ...
1 4 10 19 31 46 ...
1 5 15 35 69 121...

Rows include A005408, A005448, A005894, A008498, A008499, A008500, A008501.

Diagonals include A030662, A001700, A001791, A002054, A002694, A003516.

Square array T(n, k)=C(n+k, k)-C(n, k).

Row k has g.f. (1-x^k)/(1-x)^(k+1).

A217234 Triangle of expansion coefficients of the sum of an n X n array with equal top row and left column (extended by the rule of Pascal's triangle) in terms of the top row elements.

Original entry on oeis.org

1, 1, 4, 1, 12, 6, 1, 40, 20, 8, 1, 140, 70, 30, 10, 1, 504, 252, 112, 42, 12, 1, 1848, 924, 420, 168, 56, 14, 1, 6864, 3432, 1584, 660, 240, 72, 16, 1, 25740, 12870, 6006, 2574, 990, 330, 90, 18, 1, 97240, 48620, 22880, 10010, 4004, 1430, 440, 110, 20

Offset: 1

J. M. Bergot, Sep 28 2012

Define a finite n X n square array with indeterminate elements A(1, c), c=1..n in the top row, the same elements A(r,1 ) = A(1,r) in the first column, r=1..n, and the remaining elements defined by the Pascal triangle rule: A(r,c) = A(r,c-1)+A(r-1,c).
The triangle T(n,m) gives the coefficients in the formula Sum_{r=1..n} Sum_{c=1..n} A(r,c) = Sum_{m=1..n} T(n,m) * A(1,m).
It says how many times the first, second, third, etc. element of the first row (or the first column) contributes to the sum of the n X n array.

			1;
1,4;
1,12,6;
1,40,20,8;
1,140,70,30,10;
1,504,252,112,42,12;
1,1848,924,420,168,56,14;
1,6864,3432,1584,660,240,72,16;
1,25740,12870,6006,2574,990,330,90,18;
1,97240,48620,22880,10010,4004,1430,440,110,20;

Cf. A100320 (2nd column), A000984 (third column), A162551 (third column), A024483 (4th column), A006659 (5th column), A002058 (6th column), A030662 (row sums).

A217234_row := proc(n)
    local A,r,c,s ;
    A := array({},1..n,1..n) ;
    for r from 2 to n do
        A[r,1] := A[1,r] ;
    end do:
    for r from 2 to n do
        for c from 2 to n do
            A[r,c] := A[r,c-1]+A[r-1,c] ;
        end do:
    end do:
    s := add(add( A[r,c],c=1..n) ,r=1..n) ;
    for c from 1 to n do
        printf("%d,", coeff(s,A[1,c]) ) ;
    end do:
    return ;
end proc:
for n from 1 to 10 do
    A217234_row(n) ;
    printf(";\n") ;
end do; # R. J. Mathar, Oct 13 2012

A217234row [n_] := Module[{A, x, r, c, s }, A = Array[x, {n, n}]; Do[A[[r, 1]] = A[[1, r]], {r, 2, n}]; Do[A[[r, c]] = A[[r, c - 1]] + A[[r - 1, c]], {r, 2, n}, {c, 2, n}]; s = Sum[A[[r, c]], {r, 1, n}, {c, 1, n}]; If[n == 1, {1}, List @@ s /. x[, ] -> 1]];
Table[A217234row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 04 2023, after R. J. Mathar *)

Edited by R. J. Mathar, Oct 13 2012

Typo in data corrected by Jean-François Alcover, Nov 04 2023

cp's OEIS Frontend

A260877 Square array read by ascending antidiagonals: number of m-shape Euler numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Sage

A101475 Triangle T(n,k) read by rows: number of lattice paths from (0,0) to (0,2n) with steps (1,1) or (1,-1) that stay between the lines y=0 and y=k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A128205 a(n) = 2^(n-1)*A047240(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A172021 Start with the triangle A171661, reverse its rows, add missing powers of 2 at beginning of each row.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Derive

Maple

Mathematica

Formula

Extensions

A363570 Expansion of (1 / sqrt(1 - 8*x + 4*x^2) - 1 / (1 - x)) / 3.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A363571 Expansion of (1 / sqrt(1 - 10*x + 9*x^2) - 1 / (1 - x)) / 4.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A369335 a(n) is the sum of the numbers inside a square of side length 2n+1 located in Pascal's triangle at and below row n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A375555 Triangle read by rows: T(n, k) = abs(A181937(k, n)), where A181937 are the André numbers, for 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

A363570 Expansion of (1 / sqrt(1 - 8x + 4x^2) - 1 / (1 - x)) / 3.

A363571 Expansion of (1 / sqrt(1 - 10x + 9x^2) - 1 / (1 - x)) / 4.