A100157 -id:A100157 - cp's OEIS frontend

A213752 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

1, 6, 3, 19, 14, 5, 44, 37, 22, 7, 85, 76, 55, 30, 9, 146, 135, 108, 73, 38, 11, 231, 218, 185, 140, 91, 46, 13, 344, 329, 290, 235, 172, 109, 54, 15, 489, 472, 427, 362, 285, 204, 127, 62, 17, 670, 651, 600, 525, 434, 335, 236, 145, 70, 19, 891, 870, 813

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal:  A100157
Antidiagonal sums:  A071238
row 1,  (1,3,5,7,9,...)**(1,3,5,7,9,...): A005900
row 2,  (1,3,5,7,9,...)**(3,5,7,9,11,...): A143941
row 3,  (1,3,5,7,9,...)**(5,7,9,11,13,...): (2*k^3 + 12*k^2 + k)/6
row 4,  (1,3,5,7,9,...)**(7,9,11,13,15,,...): (2*k^3 + 18*k^2 + k)/6
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...6....19...44....85....146
3...14...37...76....135...218
5...22...55...108...185...290
7...30...73...140...235...362
9...38...91...172...285...434

Cf. A213500.

b[n_] := 2 n - 1; c[n_] := 2 n - 1;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213752 *)
Table[t[n, n], {n, 1, 40}] (* A100157 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A071238 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = 2*n - 1 + 2*x - (2*n - 3)*x^2 and g(x) = (1 - x )^4.

A329070 Array read by ascending antidiagonals: T(n, k) = (kn)!/(k^n(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x(x + 1)...*(x + n - 1) is the Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 8, 6, 1, 1, 48, 180, 24, 1, 1, 384, 12960, 8064, 120, 1, 1, 3840, 1710720, 10644480, 604800, 720, 1, 1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1, 1, 645120, 109930867200, 244635697152000, 2303884477440000, 70355755008000, 10897286400, 40320, 1

Offset: 0

Petros Hadjicostas, Nov 03 2019

For information about the function W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)) (mentioned in the Formula section below), see Theorem 3.2 in Elizalde and Noy (2003) with u = 0 and m and a in the theorem equal to our m + 1. See also the documentation of array A327722.
By using the ratio test and the Stirling approximation to the gamma function, we may show that the radius of convergence of the power series for W_m(z) is infinity (for each m >= 0). Thus, the function W_m(z) (as defined by the above power series) is entire.
If we define S(m,s) = T(n-s, s+1) for m >= 0 and 0 <= s <= m, we get the triangular array that appears in the Example section below.

			Array T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
  1,  1,     1,        1,           1,              1,  ...
  1,  2,     6,       24,         120,            720,  ...
  1,  8,   180,     8064,      604800,       68428800, ...
  1, 48, 12960, 10644480, 19813248000, 70355755008000, ...
  ...
Triangular array S(m,s) = T(m-s, s+1) (with rows m >= 0 and columns s >= 0):
  1;
  1,     1;
  1,     2,         1;
  1,     8,         6,           1;
  1,    48,       180,          24,           1;
  1,   384,     12960,        8064,         120,        1;
  1,  3840,   1710720,    10644480,      604800,      720,    1;
  1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1;
  ...

Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116).
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Wikipedia, Falling and rising factorials.

Rows include A000012 (n = 0), A000142 (n = 1), A060593 (n = 2).
Columns include A000012 (k = 1), A000165 (k = 2), A176730 (k = 3).
Ratios T(n+1,k)/(k!*T(n,k)) include A000012 (k = 1), A000027 (k = 2), A000326 (k = 3), A100157 (k = 4), A234043 (k = 5).
Cf. A111004, A117226, A202213, A327722.

A := (n, k) -> `if`(k=0, 1, (GAMMA(1/k)*GAMMA(k*n+1))/(GAMMA(n+1/k)*k^n)):
seq(seq(A(n-k-1, k), k=1..n-1), n=0..10); # Peter Luschny, Nov 04 2019

T(0,k) = 1, T(1,k) = k!, and T(2,k) = (2*k)!/(k + 1) for k >= 1.
T(n,1) = 1, T(n,2) = (2*n)!!, and T(n,3) is related to the Airy functions (see the documentation of A176730).
T(n+1,k) = (k-1)! * binomial(k*(n+1), k-1) * T(n,k) for n >= 0 and k >= 1.
T(n+1,k)/(k! * T(n,k)) = Cat(n+1, k), where Cat(d, k) = binomial(k*d, k)/(k * (d - 1) + 1) is a Fuss-Catalan number; see Theorem 1.2 in Schuetz and Whieldon (2014).
If F(k,z) = Sum_{n >= 0} z^(k*n)/T(n,k), then F(k,z) satisfies the o.d.e. F^(k-1)(k,z) - z*F(k,z) = 0.
If W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)), then 1/W_m(z) is the e.g.f. of row m of A327722(m,n), which counts permutations of [n] that avoid the consecutive pattern 12...(m+1)(m+3)(m+2) (or equivalently, the consecutive pattern (m+3)(m+2)...(3)(1)(2)).
The function W_m(z) satisfies the o.d.e. W_m^(m+2)(z) + z*W_m'(z) = 0 with W_m(0) = 1, W_m'(0) = -1, and W_m^(s)(0) = 0 for s = 2..(m + 1).

A100156 Structured truncated tetrahedral numbers.

Original entry on oeis.org

1, 12, 44, 108, 215, 376, 602, 904, 1293, 1780, 2376, 3092, 3939, 4928, 6070, 7376, 8857, 10524, 12388, 14460, 16751, 19272, 22034, 25048, 28325, 31876, 35712, 39844, 44283, 49040, 54126, 59552, 65329, 71468, 77980, 84876, 92167, 99864, 107978, 116520, 125501, 134932

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100155, A100157 for adjacent structured Archimedean solids; A100145 for more on structured polyhedral numbers. Similar to truncated tetrahedral numbers A005906.

[(1/6)*(11*n^3-3*n^2-2*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

Table[(11n^3-3n^2-2n)/6,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,12,44,108},40] (* Harvey P. Dale, Sep 28 2011 *)

vector(50, n, (11*n^3 - 3*n^2 - 2*n)/6) \\ G. C. Greubel, Oct 18 2018

a(n) = (1/6)*(11*n^3 - 3*n^2 - 2*n).
From Harvey P. Dale, Sep 28 2011: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=12, a(2)=44, a(3)=108.
G.f.: x*(2*x*(x+4)+1)/(x-1)^4. (End)
E.g.f.: x*(6 + 30*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Oct 18 2018

A100171 Structured triakis octahedral numbers (vertex structure 4).

Original entry on oeis.org

1, 14, 60, 160, 335, 606, 994, 1520, 2205, 3070, 4136, 5424, 6955, 8750, 10830, 13216, 15929, 18990, 22420, 26240, 30471, 35134, 40250, 45840, 51925, 58526, 65664, 73360, 81635, 90510, 100006, 110144

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A100157 = alternate vertex; A100145 for more on structured numbers.

[(1/6)*(21*n^3-27*n^2+12*n): n in [1..40]]; // Vincenzo Librandi, Jul 27 2011

Table[(21n^3-27n^2+12n)/6,{n,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{1,14,60,160},40] (* Harvey P. Dale, Jun 28 2011 *)

a(n)=(1/6)*(21*n^3-27*n^2+12*n).
a(0)=1, a(1)=14, a(2)=60, a(3)=160, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Jun 28 2011
G.f.: (10*x^2+10*x+1)/(x-1)^4. - Harvey P. Dale, Jun 28 2011

A258582 a(n) = n(2n + 1)(4n + 1)/3.

Original entry on oeis.org

0, 5, 30, 91, 204, 385, 650, 1015, 1496, 2109, 2870, 3795, 4900, 6201, 7714, 9455, 11440, 13685, 16206, 19019, 22140, 25585, 29370, 33511, 38024, 42925, 48230, 53955, 60116, 66729, 73810, 81375, 89440, 98021, 107134, 116795, 127020, 137825, 149226, 161239, 173880

Offset: 0

Ilya Gutkovskiy, Nov 06 2015

First bisection of the square pyramidal numbers (A000330).

Eric Weisstein's World of Mathematics, Square Pyramidal Number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A001477, A005408, A016813, A053126 (partial sums), A100157.

[n*(2*n+1)*(4*n+1)/3: n in [0..50]]; // Wesley Ivan Hurt, Nov 17 2015

A258582:=n->n*(2*n + 1)*(4*n + 1)/3: seq(A258582(n), n=0..50); # Wesley Ivan Hurt, Nov 17 2015

Table[(1/3) n (2 n + 1) (4 n + 1), {n, 0, 45}]

vector(100, n, n--; n*(2*n+1)*(4*n+1)/3) \\ Altug Alkan, Nov 06 2015

concat(0, Vec((5*x + 10*x^2 + x^3)/(1 - x)^4 + O(x^50))) \\ Altug Alkan, Nov 06 2015

G.f.: x*(5 + 10*x + x^2)/(1 - x)^4.
a(n) = A000330(2*n).
Sum_{n>0} 1/a(n) = 3*(6 - Pi - 4*log(2)) = 0.25745587...
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Nov 18 2015
a(n) = A006918(4*n-1) = A053307(4*n-1) = A228706(4*n-1) for n>0. - Bruno Berselli, Nov 18 2015
a(n) = Sum_{k=1..2*n} k^2 (see the first comment).  E.g.f. exp(x)*(5*x+ 20*x^2/2+16*x^3/3!). - Wolfdieter Lang, Mar 13 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) + 6*sqrt(2)*log(1+sqrt(2)) + 3*(sqrt(2)-1/2)*Pi - 18. - Amiram Eldar, Sep 17 2022

A316224 a(n) = n(2n + 1)(4n + 1).

Original entry on oeis.org

0, 15, 90, 273, 612, 1155, 1950, 3045, 4488, 6327, 8610, 11385, 14700, 18603, 23142, 28365, 34320, 41055, 48618, 57057, 66420, 76755, 88110, 100533, 114072, 128775, 144690, 161865, 180348, 200187, 221430, 244125, 268320, 294063, 321402, 350385, 381060, 413475, 447678, 483717

Offset: 0

Bruno Berselli, Jun 27 2018

Sums of the consecutive integers from A000384(n) to A000384(n+1)-1. This is the case s=6 of the formula n*(n*(s-2) + 1)*(n*(s-2) + 2)/2 related to s-gonal numbers.

The inverse binomial transform is 0, 15, 60, 48, 0, ... (0 continued).

			Row sums of the triangle:
|  0 |  ................................................................. 0
|  1 |  2  3  4  5  .................................................... 15
|  6 |  7  8  9 10 11 12 13 14  ........................................ 90
| 15 | 16 17 18 19 20 21 22 23 24 25 26 27  ........................... 273
| 28 | 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  ............... 612
| 45 | 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65  .. 1155
...
where:
. first column is A000384,
. second column is A130883 (without 1),
. third column is A033816,
. diagonal is A014106,
. 0, 2, 8, 18, 32, 50, ... are in A001105.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First bisection of A059270 and subsequence of A034828, A047866, A109900, A290168.

Sums of the consecutive integers from P(s,n) to P(s,n+1)-1, where P(s,k) is the k-th s-gonal number: A027480 (s=3), A055112 (s=4), A228888 (s=5).

GAP
```
List([0..40], n -> n*(2*n+1)*(4*n+1));
```

[n*(2*n+1)*(4*n+1) for n in 0:40] |> println

Magma
```
[n*(2*n+1)*(4*n+1): n in [0..40]];
```

seq(n*(2*n+1)*(4*n+1),n=0..40); # Muniru A Asiru, Jun 27 2018

Table[n (2 n + 1) (4 n + 1), {n, 0, 40}]

Maxima
```
makelist(n*(2*n+1)*(4*n+1), n, 0, 40);
```
PARI
```
vector(40, n, n--; n*(2*n+1)*(4*n+1))
```
Python
```
[n*(2*n+1)*(4*n+1) for n in range(40)]
```
Sage
```
[n*(2*n+1)*(4*n+1) for n in (0..40)]
```

O.g.f.: 3*x*(5 + 10*x + x^2)/(1 - x)^4.
E.g.f.: x*(15 + 30*x + 8*x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) =  3*A258582(n).
a(n) = -3*A100157(-n).
Sum_{n>0} 1/a(n) = 2*(3 - log(4)) - Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 2*sqrt(2)*log(1+sqrt(2)) + (sqrt(2)-1/2)*Pi - 6. - Amiram Eldar, Sep 17 2022

A227300 Rising diagonal sums of triangle of Fibonacci polynomials (rows displayed as centered text).

Original entry on oeis.org

1, 2, 2, 3, 7, 11, 16, 28, 48, 77, 126, 211, 349, 573, 947, 1568, 2588, 4271, 7058, 11661, 19256, 31804, 52538, 86779, 143329, 236744, 391046, 645900, 1066850, 1762163, 2910634, 4807590, 7940870, 13116238, 21664568, 35784145, 59105987, 97627533, 161254953, 266350689

Offset: 1

John Molokach, Jul 09 2013

nonn

Rising diagonal sums of triangle A011973, taken with rows as centered text.

			a(1)  = 1;
a(2)  = 1 +  1;
a(3)  = 1 +  1;
a(4)  = 1 +  1 +  1;
a(5)  = 1 +  1 +  3 +  2;
a(6)  = 1 +  1 +  5 +  4;
a(7)  = 1 +  1 +  7 +  6 +  1;
a(8)  = 1 +  1 +  9 +  8 +  6 +  3;
a(9)  = 1 +  1 + 11 + 10 + 15 + 10;
a(10) = 1 +  1 + 13 + 12 + 28 + 21 +  1.

Index entries for linear recurrences with constant coefficients, signature (1,0,2,0,0,-1).

Cf. A011973 (triangle), A000045 (row sums of triangle), A005314 (falling diagonal sums of triangle). Expansion of terms begin with A055624 at a(1) and adds A016813 at a(4), A016754 at a(7), and A100157 at a(10).

LinearRecurrence[{1, 0, 2, 0, 0, -1}, {1, 2, 2, 3, 7, 11}, 40] (* T. D. Noe, Jul 11 2013 *)

a(n) = if(n<=1, 1, sum(k=0, floor((n-1)/3), binomial(2*n-2-5*k,k)+binomial(2*n-1-5*k,k)) ); \\ Joerg Arndt, Jul 11 2013

a(n) = Sum_{k=0..floor((n-1)/3)} (binomial(2*n-2-5*k,k) + binomial(2*n-3-5*k,k)) for n >= 2; a(1)=1. - John Molokach, Jul 11 2013
a(n) = a(n-1) + 2*a(n-3) - a(n-6), starting with {1, 2, 2, 3, 7, 11}. - T. D. Noe, Jul 11 2013
G.f.: x*(1+x-x^3)/(1-x-2*x^3+x^6) - John Molokach, Jul 15 2013
a(n) = Sum_{k=0..floor((2n-1)/3)} binomial(2n-k-2-3*floor(k/2),floor(k/2)). - John Molokach, Jul 29 2013

A269845 Irregular triangle read by rows: T(n,k) = (k/2+1/2)^2 if odd-k otherwise T(n,k) = (n-k/2)^2 where n >= 1, k = 0..2*n-1.

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 9, 1, 4, 4, 1, 9, 16, 1, 9, 4, 4, 9, 1, 16, 25, 1, 16, 4, 9, 9, 4, 16, 1, 25, 36, 1, 25, 4, 16, 9, 9, 16, 4, 25, 1, 36, 49, 1, 36, 4, 25, 9, 16, 16, 9, 25, 4, 36, 1, 49, 64, 1, 49, 4, 36, 9, 25, 16, 16, 25, 9, 36, 4, 49, 1, 64, 81, 1, 64, 4, 49, 9, 36, 16, 25, 25, 16, 36, 9, 49, 4, 64, 1, 81, 100, 1, 81, 4, 64, 9, 49, 16, 36, 25, 25, 36, 16, 49

Offset: 1

Kival Ngaokrajang, Mar 06 2016

Inspired by A268317, but change to n+1 X n instead of Fib(n+1) X Fib(n).

There are triangles appearing along main diagonal. If the area of the smallest triangles are defined as 1, then the areas of all other triangles seem to be square numbers. Conjectures: (i) Odd terms of row sum/2 is A100157. (ii) Even terms of row sum/2 is A258582. See illustration in links.

			Irregular triangle begins:
n\k 0  1   2  3   4  5   6   7   8   9  10 11 12  13 14  15 ...
1   1, 1
2   4, 1,  1, 4
3   9, 1,  4, 4,  1, 9
4  16, 1,  9, 4,  4, 9,  1, 16
5  25, 1, 16, 4,  9, 9,  4, 16,  1, 25
6  36, 1, 25, 4, 16, 9,  9, 16,  4, 25, 1, 36
7  49, 1, 36, 4, 25, 9, 16, 16,  9, 25, 4, 36, 1, 49
8  64, 1, 49, 4, 36, 9, 25, 16, 16, 25, 9, 36, 4, 49, 1, 64
...

Kival Ngaokrajang, Illustration of initial terms, Row sum

Cf. A100157, A258582, A268317.

Table[If[OddQ@ k, (k/2 + 1/2)^2, (n - k/2)^2], {n, 8}, {k, 0, 2 n - 1}] // Flatten (* Michael De Vlieger, Apr 01 2016 *)

for (n = 1, 20, for (k = 0, 2*n-1, if (Mod(k,2)==0, t = (n-k/2)^2, t = (k/2+1/2)^2); print1(t, ", ")))

T(n,k) = (k/2+1/2)^2 if odd-k, T(n,k) = (n-k/2)^2 if even-k; n >= 1, k = 0..2*n-1.

A270309 Irregular triangle read by rows: T(n,k) = ((n-k)+1)^2 if odd-n and odd-k; T(n,k) = k^2 if odd-n and even-k; T(n,k) = (n/2-(k/2-1/2))^2 if even-n and odd-k; T(n,k) = (k/2+1)^2 if even-n and even-k; where n >= 1, k = 1..2*n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 9, 4, 1, 1, 4, 9, 4, 1, 1, 4, 4, 1, 1, 4, 25, 4, 9, 16, 1, 1, 16, 9, 4, 25, 9, 1, 4, 4, 1, 9, 9, 1, 4, 4, 1, 9, 49, 4, 25, 16, 9, 36, 1, 1, 36, 9, 16, 25, 4, 49, 16, 1, 9, 4, 4, 9, 1, 16, 16, 1, 9, 4, 4, 9, 1, 16, 81, 4, 49, 16, 25, 36, 9, 64, 1, 1, 64, 9, 36, 25, 16, 49, 4, 81, 25, 1, 16, 4, 9, 9, 4, 16, 1, 25, 25, 1, 16, 4, 9, 9, 4, 16, 1, 25

Offset: 1

Kival Ngaokrajang, Mar 15 2016

Refer to A269845, but change to n+2 X n instead of n+1 X n.

There are triangles appearing along main diagonal. If the area of the smallest triangles are defined as 1, then the areas of all other triangles seem to be square numbers. Conjectures: (i) Even terms of row sum is A002492. (ii) Odd terms of row sum/2 is A100157. See illustration in links.

			Irregular triangle begins:
n\k  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  ...
1    1,  1
2    1,  1,  1,  1
3    9,  4,  1,  1,  4,  9
4    4,  1,  1,  4,  4,  1,  1,  4
5   25,  4,  9, 16,  1,  1, 16,  9,  4, 25
6    9,  1,  4,  4,  1,  9,  9,  1,  4,  4,  1,  9
7   49,  4, 25, 16,  9, 36,  1,  1, 36,  9, 16, 25,  4, 49
8   16,  1,  9,  4,  4,  9,  1, 16, 16,  1,  9,  4,  4,  9,  1, 16
...

Kival Ngaokrajang, Illustration of initial terms, Row sum

Cf. A002492, A100157, A269845.

A137211 Generalized or s-Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 5, 12, 22, 1, 14, 55, 140, 285, 1, 42, 273, 969, 2530, 5481, 1, 132, 1428, 7084, 23751, 62832, 141778, 1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348, 1, 1430, 43263, 420732, 2330445, 9203634, 28989675, 77652024

Offset: 1

Roger L. Bagula, Mar 05 2008

nonn

From R. J. Mathar, May 04 2008: (Start)
This is a triangular section of Stanica's array of s-Catalan numbers, with rows A000108, A001764, A002293-A002296, A007556, A062994, A059968,... read along diagonals in A062993 and A070914:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ...
1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, ...
1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, ...
1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, ...
1, 1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310, 1478314266, ...
1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856, ...
1, 1, 8, 92, 1240, 18278, 285384, 4638348, 77652024, 1329890705, 23190029720, ...
1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, ...
1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840, ...
(End)
The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link for this interpretation and others), so the (k+1)-th column of Stanica's array enumerates the number of (n+1)-gon partitions of a (k*(n-1)+2)-gon. Cf. A000326 (k=3), A100157 (k=4) and A234043 (k=5). - Tom Copeland, Oct 05 2014

			{1},
{1, 1},
{1, 2, 3},
{1, 5, 12, 22},
{1, 14, 55, 140, 285},
{1, 42, 273, 969, 2530, 5481},
{1, 132, 1428, 7084, 23751, 62832, 141778},
{1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348}

Heinrich Niederhausen, Catalan Traffic at the Beach, Electronic Journal of Combinatorics, Volume 9 (2002), #R33.
A. Regev, The Central Component of a Triangulation, J. Int. Seq. 16 (2013) #13.4.1
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
P. Stanica, p^q-Catalan numbers and squarefree binomial coefficients, J. Numb. Theory 100 (2003) 203-216.

t[n_, m_] := Binomial[m*n, n]/((m - 1)*n + 1); a = Table[Table[t[n, m], {m, 1, n + 1}], {n, 0, 10}]; Flatten[a]

T(n,m) = binomial(m*n,n)/((m-1)*n+1).

Edited by N. J. A. Sloane, May 16 2008

cp's OEIS Frontend

A213752 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.

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A329070 Array read by ascending antidiagonals: T(n, k) = (k*n)!/(k^n*(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol.

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A100156 Structured truncated tetrahedral numbers.

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A100171 Structured triakis octahedral numbers (vertex structure 4).

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A258582 a(n) = n*(2*n + 1)*(4*n + 1)/3.

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A316224 a(n) = n*(2*n + 1)*(4*n + 1).

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A227300 Rising diagonal sums of triangle of Fibonacci polynomials (rows displayed as centered text).

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A329070 Array read by ascending antidiagonals: T(n, k) = (kn)!/(k^n(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x(x + 1)...*(x + n - 1) is the Pochhammer symbol.

A258582 a(n) = n(2n + 1)(4n + 1)/3.

A316224 a(n) = n(2n + 1)(4n + 1).