A055998 -id:A055998 - cp's OEIS frontend

A212331 a(n) = 5n(n+5)/2.

0, 15, 35, 60, 90, 125, 165, 210, 260, 315, 375, 440, 510, 585, 665, 750, 840, 935, 1035, 1140, 1250, 1365, 1485, 1610, 1740, 1875, 2015, 2160, 2310, 2465, 2625, 2790, 2960, 3135, 3315, 3500, 3690, 3885, 4085, 4290, 4500, 4715, 4935, 5160, 5390, 5625, 5865

Offset: 0

Bruno Berselli, May 30 2012

Numbers of the form n*t(n+5,h)-(n+5)*t(n,h), where t(k,h) = k*(k+2*h+1)/2 for any h. Likewise:
A000217(n) = n*t(n+1,h)-(n+1)*t(n,h),
A005563(n) = n*t(n+2,h)-(n+2)*t(n,h),
A140091(n) = n*t(n+3,h)-(n+3)*t(n,h),
A067728(n) = n*t(n+4,h)-(n+4)*t(n,h) (n>0),
A140681(n) = n*t(n+6,h)-(n+6)*t(n,h).
This is the case r=7 in the formula:
u(r,n) = (P(r, P(n+r, r+6)) - P(n+r, P(r, r+6))) / ((r+5)*(r+6)/2)^2, where P(s, m) is the m-th s-gonal number.
Also, a(k) is a square for k = (5/2)*(A078986(n)-1).
Sum of reciprocals of a(n), for n>0: 137/750.
Also, numbers h such that 8*h/5+25 is a square.
The table given below as example gives the dimensions D(h, n) of the irreducible SU(3) multiplets (h,n). See the triangle A098737 with offset 0, and the comments there, also with a link and the Coleman reference. - Wolfdieter Lang, Dec 18 2020

			From the first and second comment derives the following table:
----------------------------------------------------------------
h \ n | 0   1    2    3    4    5    6    7    8    9    10
------|---------------------------------------------------------
0     | 0,  1,   3,   6,  10,  15,  21,  28,  36,  45,   55, ...  (A000217)
1     | 0,  3,   8,  15,  24,  35,  48,  63,  80,  99,  120, ...  (A005563)
2     | 0,  6,  15,  27,  42,  60,  81, 105, 132, 162,  195, ...  (A140091)
3     | 0, 10,  24,  42,  64,  90, 120, 154, 192, 234,  280, ...  (A067728)
4     | 0, 15,  35,  60,  90, 125, 165, 210, 260, 315,  375, ...  (A212331)
5     | 0, 21,  48,  81, 120, 165, 216, 273, 336, 405,  480, ...  (A140681)
6     | 0, 28,  63, 105, 154, 210, 273, 343, 420, 504,  595, ...
7     | 0, 36,  80, 132, 192, 260, 336, 420, 512, 612,  720, ...
8     | 0, 45,  99, 162, 234, 315, 405, 504, 612, 729,  855, ...
9     | 0, 55, 120, 195, 280, 375, 480, 595, 720, 855, 1000, ...
with the formula n*(h+1)*(h+n+1)/2. See also A098737.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000537, A005563, A055998, A067728, A098737, A140091, A140681.

Magma
```
[5*n*(n+5)/2: n in [0..46]];
```
Mathematica
```
Table[(5/2) n (n + 5), {n, 0, 46}]
```

a(n)=5*n*(n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 5*x*(3-2*x)/(1-x)^3.
a(n) = a(-n-5) = 5*A055998(n).
E.g.f.: (5/2)*x*(x + 6)*exp(x). - G. C. Greubel, Jul 21 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/25 - 47/750. - Amiram Eldar, Feb 26 2022

Extended by Bruno Berselli, Aug 05 2015

A350248 Triangle read by rows: T(n,k) is the number of noncrossing partitions of an n-set into k blocks of size 3 or more, n >= 0, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 3, 0, 1, 7, 0, 1, 12, 0, 1, 18, 12, 0, 1, 25, 45, 0, 1, 33, 110, 0, 1, 42, 220, 55, 0, 1, 52, 390, 286, 0, 1, 63, 637, 910, 0, 1, 75, 980, 2275, 273, 0, 1, 88, 1440, 4900, 1820, 0, 1, 102, 2040, 9520, 7140, 0, 1, 117, 2805, 17136, 21420, 1428

Offset: 0

Andrew Howroyd and Janaka Rodrigo, Dec 21 2021

			Triangle begins:
  1;
  0;
  0;
  0, 1;
  0, 1;
  0, 1;
  0, 1,   3;
  0, 1,   7;
  0, 1,  12;
  0, 1,  18,   12;
  0, 1,  25,   45;
  0, 1,  33,  110;
  0, 1,  42,  220,   55;
  0, 1,  52,  390,  286;
  0, 1,  63,  637,  910;
  0, 1,  75,  980, 2275,  273;
  0, 1,  88, 1440, 4900, 1820;
  0, 1, 102, 2040, 9520, 7140;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1750 (rows 0..100)

Columns k=2..5 are A055998, A350116, A350286, A350303.
Row sums are A114997.
Cf. A001263 (blocks of any size), A108263 (blocks of size 2 or more).

T(n)={my(p=1+O(x^3)); for(i=1, n\3, p=1+y*(x*p)^3/(1-x*p)); [Vecrev(t)| t<-Vec(p + O(x*x^n))]}
{my(A=T(12)); for(i=1, #A, print(A[i]))}

T(n,k) = if(n==0 || k>n\3, k==0, binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1)) \\ Andrew Howroyd, Dec 31 2021

G.f.: A(x,y) satisfies A(x,y) = 1 + y*(x*A(x,y))^3/(1 - x*A(x,y)).

T(n,k) = binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1) for n > 0.

A133709 Triangle read by rows: T(m,l) = number of labeled covers of size l of a finite set of m unlabeled elements (m >= 1, 1 <= l <= 2^m - 1).

Original entry on oeis.org

1, 1, 3, 3, 1, 7, 35, 140, 420, 840, 840, 1, 12, 131, 1435, 15225, 150570, 1351770, 10810800, 75675600, 454053600, 2270268000, 9081072000, 27243216000, 54486432000, 54486432000, 1, 18, 347, 7693, 185031, 4568046, 111793710, 2661422400

Offset: 1

N. J. A. Sloane, Dec 30 2007

			Triangle begins:
1
1 3 3
1 7 35 140 420 840 840
1 12 131 1435 15225 150570 1351770

A. P. Burger and J. H. van Vuuren, Balanced minimal covers of a finite set, Discr. Math. 307 (2007), 2853-2860.

Columns are given by A055998, A133710, A133711, A133712.

A133709 := proc(m,l)
        option remember;
        if l = 1 then
                1;
        else
                add((-1)^i*binomial(l,i)*binomial(2^(l-i)+m-2,m),i=0..l-1)
                - add(combinat[stirling2](l,i)*procname(m,i),i=1..l-1) ;
        end if;
end proc:
seq(seq(A133709(m,l),l=1..2^m-1),m=1..5) ; # R. J. Mathar, Nov 23 2011

T[m_, l_] := T[m, l] = If[l == 1, 1, Sum[(-1)^i Binomial[l, i] Binomial[ 2^(l-i)+m-2, m], {i, 0, l-1}] - Sum[StirlingS2[l, i] T[m, i], {i, 1, l-1} ] ];
Table[T[m, l], {m, 1, 5}, {l, 1, 2^m-1}] // Flatten (* Jean-François Alcover, Apr 01 2020, from Maple *)

Burger and van Vuuren give an explicit formula.

A009056 Numbers >= 3.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

David W. Wilson

Same as Pisot sequences E(3,4), P(3,4), T(3,4). See A008776 for definitions of Pisot sequences.
Non-Fermat-exponents, positive integers n such that there are no solutions in positive integers of the equation a^n + b^n = c^n. - Tanya Khovanova, Jul 09 2011
Sums of twin primes. - Charles R Greathouse IV, Jun 21 2012

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A009005, A020739, A055998.

Range[3,80] (* Harvey P. Dale, Sep 26 2017 *)

a(n)=n+2 \\ Charles R Greathouse IV, Jun 21 2012

a(n) = n + 2.
From R. J. Mathar, May 26 2008: (Start)
O.g.f.: x*(3-2*x)/(1-x)^2.
a(n) = A009005(n-1), n > 2. (End)
From Elmo R. Oliveira, Oct 31 2024: (Start)
E.g.f.: exp(x)*(x + 2) - 2.
a(n) = 2*a(n-1) - a(n-2) for n > 2.
a(n) = A055998(n) - A055998(n-1) = A020739(n-1)/2. (End)

A085771 Triangle read by rows. T(n, k) = A059438(n, k) for 1 <= k <= n, and T(n, 0) = n^0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 7, 3, 1, 0, 71, 32, 12, 4, 1, 0, 461, 177, 58, 18, 5, 1, 0, 3447, 1142, 327, 92, 25, 6, 1, 0, 29093, 8411, 2109, 531, 135, 33, 7, 1, 0, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1, 0, 2829325, 642581, 125316, 24892, 5226, 1146, 252, 52, 9, 1

Offset: 0

Philippe Deléham, Jul 22 2003

The convolution triangle of A003319, the number of irreducible permutations. - Peter Luschny, Oct 09 2022

			Triangle starts:
[0] [1]
[1] [0,      1]
[2] [0,      1,     1]
[3] [0,      3,     2,     1]
[4] [0,     13,     7,     3,    1]
[5] [0,     71,    32,    12,    4,   1]
[6] [0,    461,   177,    58,   18,   5,   1]
[7] [0,   3447,  1142,   327,   92,  25,   6,  1]
[8] [0,  29093,  8411,  2109,  531, 135,  33,  7, 1]
[9] [0, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1]

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).

FindStat - Combinatorial Statistic Finder, The decomposition number of a permutation.

T(2*n, n) = A308650(n).
Variants: A059439, A263484 (row reversed).
Cf. A003319, A059440, A055998, A059438, A000007, A084938.

# Uses function PMatrix from A357368.
PMatrix(10, A003319); # Peter Luschny, Oct 09 2022

# Using function delehamdelta from A084938.
def A085771_triangle(n) :
    a = [0, 1] + [(i + 3) // 2 for i in range(1, n-1)]
    b = [0^i for i in range(n)]
    return delehamdelta(a, b)
A085771_triangle(9) # Peter Luschny, Sep 10 2022

Let f(x) = Sum_{n>=0} n!*x^n, g(x) = 1 - 1/f(x). Then g(x) is the g.f. of the second column, A003319.
Triangle T(n, k) read by rows, given by [0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938.
G.f.: 1/(1 - xy/(1 - x/(1 - 2x/(1 - 2x/(1 - 3x/(1 - 3x/(1 - 4x/(1-.... (continued fraction). - Paul Barry, Jan 29 2009

A165157 Zero followed by partial sums of A133622.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 12, 13, 18, 19, 25, 26, 33, 34, 42, 43, 52, 53, 63, 64, 75, 76, 88, 89, 102, 103, 117, 118, 133, 134, 150, 151, 168, 169, 187, 188, 207, 208, 228, 229, 250, 251, 273, 274, 297, 298, 322, 323, 348, 349, 375, 376, 403, 404, 432, 433, 462, 463, 493, 494, 525

Offset: 0

Jaroslav Krizek, Sep 05 2009

A133622 is a toothed sequence.

Interleaving of A055998 and A034856.

			From _Stefano Spezia_, Jul 10 2020: (Start)
Illustration of the initial terms for n > 0:
o    o      o      o         o        o
     o o    o o    o o       o o      o o
            o      o         o        o
                   o o o     o o o    o o o
                             o        o
                                      o o o o
(1)  (3)   (4)    (7)       (8)      (12)
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Equals -1+A101881.
a(n) = A117142(n+2)-2 = A055802(n+6)-3 = A114220(n+5)-3 = A134519(n+3)-3.
Cf. A133622, A055998, A034856.

a165157 n = a165157_list !! n
a165157_list = scanl (+) 0 a133622_list
-- Reinhard Zumkeller, Feb 20 2015

m:=60; T:=[ 1+(1+(-1)^n)*n/4: n in [1..m] ]; [0] cat [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..m] ]; // Klaus Brockhaus, Sep 06 2009

[ n le 2 select n-1 else n le 4 select n else 2*Self(n-2)-Self(n-4)+1: n in [1..61] ]; // Klaus Brockhaus, Sep 06 2009

a(0) = 0, a(2*n) = a(2*n-1) + n + 1, a(2*n+1) = a(2*n) + 1.
a(n) = (n^2+10*n)/8 if n is even, a(n) = (n^2+8*n-1)/8 if n is odd.
a(2*k) = A055998(k) = k*(k+5)/2; a(2*k+1) = A034856(k+1) = k*(k+5)/2+1.
a(n) = 2*a(n-2)-a(n-4)+1 for n > 3; a(0)=0, a(1)=1, a(2)=3, a(3)=4. - Klaus Brockhaus, Sep 06 2009
a(n) = (2*n*(n+9)-1+(2*n+1)*(-1)^n)/16. - Klaus Brockhaus, Sep 06 2009
a(n) = n+binomial(1+floor(n/2),2). - Mircea Merca, Feb 18 2012
G.f.: x*(1+2*x-x^2-x^3)/((1-x)^3*(1+x)^2). - Klaus Brockhaus, Sep 06 2009
From Stefano Spezia, Jul 10 2020: (Start)
E.g.f.: (x*(9 + x)*cosh(x) + (-1 + 11*x + x^2)*sinh(x))/8.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. (End)

Edited and extended by Klaus Brockhaus, Sep 06 2009

A196842 Table of the elementary symmetric functions a_k(1,2,4,5,...,n+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 7, 14, 8, 1, 12, 49, 78, 40, 1, 18, 121, 372, 508, 240, 1, 25, 247, 1219, 3112, 3796, 1680, 1, 33, 447, 3195, 12864, 28692, 32048, 13440, 1, 42, 744, 7218, 41619, 144468, 290276, 301872, 120960, 1, 52, 1164, 14658, 113799, 560658, 1734956, 3204632, 3139680, 1209600

Offset: 0

Wolfdieter Lang, Oct 24 2011

For the symmetric functions a_k and the definition of the triangles S_j(n,k) see a comment in A196841. Here x[1]=1, x[2]=2, and x[j]=j+1 for j=3,...,n. This is the triangle S_3(n,k), n>=0, k=0..n. The first three rows coincide with those of triangle A094638.

			n\k   0    1    2     3      4      5     6       7  ...
0:    1
1:    1    1
2:    1    3    2
3:    1    7   14     8
4:    1   12   49    78     40
5:    1   18  121   372    508    240
6:    1   25  247  1219   3112   3796   1680
7:    1   33  447  3195  12864  28692  32048  13440
...
a(1,0) = a_0(1):= 1, a(1,1) = a_1(1)= 1.
a(3,2) = a_2(1,2,4) = 1*2 + 1*4 + 2*4 = 14.
a(3,2) = 1*|s(5,3)| - 3*|s(5,4)| + 9*|s(5,5)| = 1*35-3*10+9*1 = 14.

Cf. A094638, A145324,|A123319|, A196841, A055998 (column k=1), A002301 (diagonal), A277132 (subdiagonal).

A196842 := proc(n,k)
    if n = 1 and k =1 then
        1 ;
    else
        add( abs( combinat[stirling1](n+2,n+2-k+m))*(-3)^m,m=0..k) ;
    end if;
end proc: # R. J. Mathar, Oct 01 2016

a[n_, k_] := If[n == 1 && k == 1, 1, Sum[(-3)^m Abs[StirlingS1[n + 2, n + 2 - k + m]], {m, 0, k}]];
Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 16 2023, after R. J. Mathar *)

a(n,k) = a_k(1,2,..,n) if 0<=n<3, and a_k(1,2,4,5,...,n+1) if n>=3, with the elementary symmetric functions a_k defined in a comment to A196841.

a(n,k) = 0 if n=3, with the Stirling numbers of the first kind s(n,m)=A048994(n,m).

A209293 Inverse permutation of A185180.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 9, 7, 10, 13, 12, 14, 11, 15, 18, 19, 17, 20, 16, 21, 25, 24, 26, 23, 27, 22, 28, 32, 33, 31, 34, 30, 35, 29, 36, 41, 40, 42, 39, 43, 38, 44, 37, 45, 50, 51, 49, 52, 48, 53, 47, 54, 46, 55, 61, 60, 62, 59, 63, 58, 64, 57, 65, 56, 66, 72, 73, 71, 74, 70, 75, 69, 76, 68, 77, 67

Offset: 1

Boris Putievskiy, Jan 16 2013

Permutation of the natural numbers. a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k) by diagonals. The order of the list
if n is odd  - T(n-1,2),T(n-3,4),...,T(2,n-1),T(1,n),T(3,n-2),...T(n,1).
if n is even - T(n-1,2),T(n-3,4),...,T(3,n-2),T(1,n),T(2,n-1),...T(n,1).
Table T(n,k) contains:
Column number 1 A000217,
column number 2 A000124,
column number 3 A000096,
column number 4 A152948,
column number 5 A034856,
column number 6 A152950,
column number 7 A055998.
Row    number 1 A000982,
row    number 2 A097063.

			The start of the sequence as table:
  1....2...5...8..13..18...25...32...41...
  3....4...9..12..19..24...33...40...51...
  6....7..14..17..26..31...42...49...62...
  10..11..20..23..34..39...52...59...74...
  15..16..27..30..43..48...63...70...87...
  21..22..35..38..53..58...75...82..101...
  28..29..44..47..64..69...88...95..116...
  36..37..54..57..76..81..102..109..132...
  45..46..65..68..89..94..117..124..149...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  2,3;
  5,4,6;
  8,9,7,10;
  13,12,14,11,15;
  18,19,17,20,16,21;
  25,24,26,23,27,22,28;
  32,33,31,34,30,35,29,36;
  41,40,42,39,43,38,44,37,45;
  . . .
Row number r contains permutation from r numbers:
if r is odd  ceiling(r^2/2), ceiling(r^2/2)+1, ceiling(r^2/2)-1, ceiling(r^2/2)+2, ceiling(r^2/2)-2,...r*(r+1)/2;
if r is even ceiling(r^2/2), ceiling(r^2/2)-1, ceiling(r^2/2)+1, ceiling(r^2/2)-2, ceiling(r^2/2)+2,...r*(r+1)/2;

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

max = 10; row[n_] := Table[Ceiling[(n + k - 1)^2/2] + If[OddQ[k], 1, -1]*Floor[n/2], {k, 1, max}]; t = Table[row[n], {n, 1, max}]; Table[t[[n - k + 1, k]], {n, 1, max}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 17 2013 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
m1=int((i+j)/2)+int(i/2)*(-1)**(i+t+1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(i+t)
m=(m1+m2-1)*(m1+m2-2)/2+m1

As table T(n,k) read by antidiagonals
T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
As linear sequence
a(n) = (m1+m2-1)*(m1+m2-2)/2+m1, where
m1 = int((i+j)/2)+int(i/2)*(-1)^(i+t+1),
m2 = int((i+j+1)/2)+int(i/2)*(-1)^(i+t),
t = int((math.sqrt(8*n-7) - 1)/ 2),
i = n-t*(t+1)/2,
j = (t*t+3*t+4)/2-n.

A024191 [ (3rd elementary symmetric function of 3,4,...,n+4)/(3+4+...+n+4) ].

Original entry on oeis.org

5, 19, 47, 95, 170, 280, 434, 642, 915, 1265, 1705, 2249, 2912, 3710, 4660, 5780, 7089, 8607, 10355, 12355, 14630, 17204, 20102, 23350, 26975, 31005, 35469, 40397, 45820, 51770, 58280, 65384, 73117, 81515, 90615, 100455, 111074, 122512, 134810, 148010

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A115127.

Partial sums of A005586.

Table[n(n+1)(n^2+13n+46)/24,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{5,19,47,95,170},40] (* Harvey P. Dale, Apr 28 2014 *)
CoefficientList[Series[(5 - 6 x + 2 x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 28 2014 *)

a(n) = n*(n+1)*(n^2+13*n+46)/24 \\ Charles R Greathouse IV, Oct 21 2022

a(n)=A115127(n+2, 3), n>=2.
a(n) = n*(n+1)*(n^2+13n+46)/24 =a(n-1)+A005586(n). - Henry Bottomley, Oct 25 2001
G.f.: x*(5-6*x+2*x^2)/(1-x)^5.
a(n) = floor(A024184(n)/A055998(n+2)). -  R. J. Mathar, Sep 15 2009
a(1)=5, a(2)=19, a(3)=47, a(4)=95, a(5)=170, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Apr 28 2014

A111106 Riordan array (1, x*g(x)) where g(x) is g.f. of double factorials A001147.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 7, 3, 1, 0, 105, 36, 12, 4, 1, 0, 945, 249, 64, 18, 5, 1, 0, 10395, 2190, 441, 100, 25, 6, 1, 0, 135135, 23535, 3807, 691, 145, 33, 7, 1, 0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1

Offset: 0

Philippe Deléham, Oct 13 2005, Dec 20 2008

Triangle T(n,k), 0 <= k <= n, given by [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Rows begin:
  1;
  0,       1;
  0,       1,      1;
  0,       3,      2,     1;
  0,      15,      7,     3,    1;
  0,     105,     36,    12,    4,    1;
  0,     945,    249,    64,   18,    5,   1;
  0,   10395,   2190,   441,  100,   25,   6,  1:
  0,  135135,  23535,  3807,  691,  145,  33,  7, 1;
  0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1;

Columns: A000007, A001147, A034430; diagonals: A000012, A001477, A055998.

Cf. A084938, A111088, A112934, A168441.

# Uses function PMatrix from A357368.
PMatrix(10, n -> doublefactorial(2*n-3)); # Peter Luschny, Oct 19 2022

T(n, k) = Sum_{j=0..n-k} T(n-1, k-1+j)*A111088(j).
Sum_{k=0..n} T(n, k) = A112934(n).
G.f.: 1/(1-xy/(1-x/(1-2x/(1-3x/(1-4x/(1-... (continued fraction). - Paul Barry, Jan 29 2009
Sum_{k=0..n} T(n,k)*2^(n-k) = A168441(n). - Philippe Deléham, Nov 28 2009

cp's OEIS Frontend

A212331 a(n) = 5*n*(n+5)/2.

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A350248 Triangle read by rows: T(n,k) is the number of noncrossing partitions of an n-set into k blocks of size 3 or more, n >= 0, 0 <= k <= floor(n/3).

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A133709 Triangle read by rows: T(m,l) = number of labeled covers of size l of a finite set of m unlabeled elements (m >= 1, 1 <= l <= 2^m - 1).

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Maple

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A009056 Numbers >= 3.

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A085771 Triangle read by rows. T(n, k) = A059438(n, k) for 1 <= k <= n, and T(n, 0) = n^0.

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A165157 Zero followed by partial sums of A133622.

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Magma

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A196842 Table of the elementary symmetric functions a_k(1,2,4,5,...,n+1).

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A212331 a(n) = 5n(n+5)/2.