A001296 -id:A001296 - cp's OEIS frontend

A254142 a(n) = (9n+10)binomial(n+9,9)/10.

1, 19, 154, 814, 3289, 11011, 32032, 83512, 199342, 442442, 923780, 1830764, 3468374, 6317234, 11113784, 18958808, 31461815, 50930165, 80613390, 125014890, 190285095, 284712285, 419329560, 608658960, 871616460, 1232604516, 1722822024, 2381824984

Offset: 0

Bruno Berselli, Jan 26 2015

Partial sums of A056003.

If n is of the form 8*k+2*(-1)^k-1 or 8*k+2*(-1)^k-2 then a(n) is odd.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000581, A001287, A056003.

Cf. sequences of the type (k*n+k+1)*binomial(n+k,k)/(k+1): A000217 (k=1), A000330 (k=2), A001296 (k=3), A034263 (k=4), A051946 (k=5), A034265 (k=6), A034266 (k=7), A056122 (k=8), this sequence (k=9).

List([0..30], n-> (9*n+10)*Binomial(n+9,9)/10); # G. C. Greubel, Aug 28 2019

[(9*n+10)*Binomial(n+9,9)/10: n in [0..30]];

seq((9*n+10)*binomial(n+9,9)/10, n=0..30); # G. C. Greubel, Aug 28 2019

Table[(9n+10)Binomial[n+9, 9]/10, {n, 0, 30}]

vector(30, n, n--; (9*n+10)*binomial(n+9, 9)/10)

[(9*n+10)*binomial(n+9,9)/10 for n in (0..30)]

G.f.: (1 + 8*x)/(1-x)^11.
a(n) = Sum_{i=0..n} (i+1)*A000581(i+8).
a(n+1) = 8*A001287(n+10) + A001287(n+11).

A005460 a(n) = (3n+4)(n+3)!/24.

Original entry on oeis.org

1, 7, 50, 390, 3360, 31920, 332640, 3780000, 46569600, 618710400, 8821612800, 134399865600, 2179457280000, 37486665216000, 681734237184000, 13071512982528000, 263564384219136000, 5575400435404800000, 123469776914964480000, 2856835183101419520000

Offset: 0

N. J. A. Sloane

Essentially Stirling numbers of second kind - third external diagonal of Worpitzky triangle A028246.

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
John K. Sikora, On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles, arXiv:1806.00887 [math.NT], 2018.

Cf. A028246.

[(3*n+4)*Factorial(n+3)/24: n in [0..20]]; // Vincenzo Librandi, Oct 08 2011

Table[StirlingS2[n+3, n+1]*n!, {n,0,20}]

a(n)=(3*n+4)*(n+3)!/24 \\ Charles R Greathouse IV, Jun 30 2017

[factorial(n)*stirling_number2(n+3,n+1) for n in range(21)] # G. C. Greubel, Nov 22 2022

E.g.f.: (1+2*x)/(1-x)^5.

a(n) = S2(n+3, n+1)*n! = n!*A001296(n+1). - Olivier Gérard, Sep 13 2016

A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 7, 15, 9, 1, 10, 36, 54, 27, 1, 13, 66, 162, 189, 81, 1, 16, 105, 360, 675, 648, 243, 1, 19, 153, 675, 1755, 2673, 2187, 729, 1, 22, 210, 1134, 3780, 7938, 10206, 7290, 2187, 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561, 1, 28, 351, 2592, 12474, 40824, 91854, 139968, 137781, 78732, 19683

Offset: 0

N. J. A. Sloane, May 03 2000

Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005
Triangle read by rows, n-th row = X^(n-1) * [1, 1, 0, 0, 0, ...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (3,3,3,...) in the subdiagonal; given row 0 = 1. - Gary W. Adamson, Jul 19 2008
Fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(2x+1)^n; see A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. - Clark Kimberling, Aug 04 2011

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  7,  15,   9;
  1, 10,  36,  54,   27;
  1, 13,  66, 162,  189,   81;
  1, 16, 105, 360,  675,  648,  243;
  1, 19, 153, 675, 1755, 2673, 2187, 729;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

Cf. A000007, A000012, A001296, A006138, A006234, A016777, A024462.
Cf. A051836, A062741, A080420, A080421, A080422, A080423, A081294.
Cf. A084938, A098399, A110523, A133494, A136158, A193722.

A038763:= func< n,k | n eq 0 select 1 else 3^(k-1)*(3*n-2*k)*Binomial(n,k)/n >;
[A038763(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 27 2023

A038763[n_,k_]:= If[n==0, 1, 3^(k-1)*(3*n-2*k)*Binomial[n,k]/n];
Table[A038763[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 27 2023 *)

T(n,k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); T(n-1,k) + 3*T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jul 25 2023

def A038763(n,k): return 1 if (n==0) else 3^(k-1)*(3*n-2*k)*binomial(n,k)/n
flatten([[A038763(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 27 2023

T(n, 0)=1; T(1, 1)=1; T(n, k)=0 for k>n; T(n, k) = T(n-1, k-1)*3 + T(n-1, k) for n >= 2.
Sum_{k=0..n} T(n,k) = A081294(n). - Philippe Deléham, Sep 22 2006
T(n, k) = A136158(n, n-k). - Philippe Deléham, Dec 17 2007
G.f.: (1-2*x*y)/(1-(3*y+1)*x). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 27 2023: (Start)
T(n, 0) = A000012(n).
T(n, 1) = A016777(n-1).
T(n, 2) = A062741(n-1).
T(n, 3) = 9*A002411(n-2).
T(n, 4) = 27*A001296(n-3).
T(n, 5) = 81*A051836(n-4).
T(n, n) = A133494(n).
T(n, n-1) = A006234(n+2).
T(n, n-2) = A080420(n-2).
T(n, n-3) = A080421(n-3).
T(n, n-4) = A080422(n-4).
T(n, n-5) = A080423(n-5).
T(2*n, n) = 4*A098399(n-1) + (2/3)*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006138(n-1) + (2/3)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A110523(n-1) + (4/3)*[n=0]. (End)

More terms from Michel Marcus, Jul 25 2023

A213086 Square array read by antidiagonals: T(n,m) (n>=1, m>=0) is the number of partitions of mn that are the sum of m not necessarily distinct partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 4, 1, 1, 7, 14, 10, 5, 1, 1, 11, 25, 30, 15, 6, 1, 1, 15, 53, 65, 55, 21, 7, 1, 1, 22, 89, 173, 140, 91, 28, 8, 1, 1, 30, 167, 343, 448, 266, 140, 36, 9, 1, 1, 42, 278, 778, 1022, 994, 462, 204, 45, 10, 1, 1, 56, 480, 1518, 2710, 2562, 1974, 750, 285, 55, 11, 1

Offset: 1

N. J. A. Sloane, Jun 05 2012

			The array begins:
  1,  1,   1,   1,    1,    1,     1,     1,     1,      1, ...
  1,  2,   3,   4,    5,    6,     7,     8,     9,     10, ...
  1,  3,   6,  10,   15,   21,    28,    36,    45,     55, ...
  1,  5,  14,  30,   55,   91,   140,   204,   285,    385, ...
  1,  7,  25,  65,  140,  266,   462,   750,  1155,   1705, ...
  1, 11,  53, 173,  448,  994,  1974,  3606,  6171,  10021, ...
  1, 15,  89, 343, 1022, 2562,  5670, 11418, 21351,  37609, ...
  1, 22, 167, 778, 2710, 7764, 19314, 43164, 88671, 170170, ...
  ...

Alois P. Heinz, Antidiagonals n = 1..18, flattened
N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.

Columns are A000041, A002219, A002220, A002221, A002222. Cf. A213074.
Rows are A000027, A000217, A000330, A001296, A207361.
Main diagonal gives A284645.

with(combinat):
g:= proc(n, m) option remember;
      `if`(m>1, map(x-> map(y-> sort([x[], y[]]), g(n, 1))[],
       g(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x),
       {partition(n)[]}), {[]}))
    end:
T:= (n, m)-> nops(g(n, m)):
seq(seq(T(d-m, m), m=0..d-1), d=1..12);  # Alois P. Heinz, Jul 11 2012

T[n_, m_] := Module[{ip, lg, i}, ip = IntegerPartitions[n]; lg = Length[ ip]; i[0]=1; Table[Join[Sequence @@ Table[ip[[i[k]]], {k, 1, m}]] // Sort, Evaluate[Sequence @@ Table[{i[k], i[k-1], lg}, {k, 1, m}]]] // Flatten[#, m-1]& // Union // Length]; T[_, 0] = 1;
Table[T[n-m, m], {n, 1, 12}, {m, 0, n - 1}] // Flatten (* Jean-François Alcover, May 25 2016 *)

Row n is a polynomial in m: see A213074 for the coefficients.

More terms and cross-references from Alois P. Heinz, Jul 11 2012

A037960 a(n) = n(3n+1)*(n+2)!/24.

Original entry on oeis.org

0, 1, 14, 150, 1560, 16800, 191520, 2328480, 30240000, 419126400, 6187104000, 97037740800, 1612798387200, 28332944640000, 524813313024000, 10226013557760000, 209144207720448000, 4480594531725312000, 100357207837286400000, 2345925761384325120000, 57136703662028390400000

Offset: 0

N. J. A. Sloane

For n>=1, a(n) is equal to the number of surjections from {1,2,..,n+2} onto {1,2,...,n}. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007

Identity (1.18) in H. W. Gould, Combinatorial Identities, Morgantown, 1972; page 3.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
H. W. Gould, ed. J. Quaintance, Combinatorial Identities, May 2010 (identity 10.3, p.45)

Cf. A000142, A001286, A001296, A019538, A037960, A131689.

Cf. A037959, A037961, A037962, A037963.

[Factorial(n+2)*n*(3*n+1)/24: n in [0..25]]; // Vincenzo Librandi, Feb 20 2017

Table[(n+2)!*n*(3n+1)/24,{n,0,20}] (* Harvey P. Dale, Oct 16 2014 *)

n*(3*n+1)*(n+2)!/24 \\ Charles R Greathouse IV, Nov 02 2011

[factorial(n)*stirling_number2(n+2, n) for n in (0..30)] # G. C. Greubel, Jun 20 2022

a(n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - Vladimir Kruchinin, Jun 01 2013
(3*n-2)*(n-1)*a(n) - n*(n+2)*(3*n+1)*a(n-1) = 0. - R. J. Mathar, Jul 26 2015
E.g.f.: x*(1 + 2*x)/(1 - x)^5. - Ilya Gutkovskiy, Feb 20 2017
From G. C. Greubel, Jun 20 2022: (Start)
a(n) = n!*StirlingS2(n+2, n).
a(n) = A131689(n+2, n).
a(n) = A019538(n+2, n). (End)

More terms from Vincenzo Librandi, Feb 20 2017

A082652 Triangle read by rows: T(n,k) is the number of squares that can be found in a k X n rectangular grid of little squares, for 1 <= k <= n.

Original entry on oeis.org

1, 2, 5, 3, 8, 14, 4, 11, 20, 30, 5, 14, 26, 40, 55, 6, 17, 32, 50, 70, 91, 7, 20, 38, 60, 85, 112, 140, 8, 23, 44, 70, 100, 133, 168, 204, 9, 26, 50, 80, 115, 154, 196, 240, 285, 10, 29, 56, 90, 130, 175, 224, 276, 330, 385, 11, 32, 62, 100, 145, 196, 252, 312, 375, 440, 506

Offset: 1

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), May 16 2003

Here the squares being counted have sides parallel to the gridlines; for all squares, see A130684.
T(n,k) also is the total number of balls in a pyramid of balls on an n X k rectangular base. - N. J. A. Sloane, Nov 17 2007. For example, if the base is 4 X 2, the total number of balls is 4*2 + 3*1 = 11 = T(4,2).
Row sums give A001296. - Vincenzo Librandi Mar 26 2019

			Let X represent a small square. Then T(3,2) = 8 because here
  XXX
  XXX
we can see 8 squares, 6 of side 1, 2 of side 2.
Triangle begins:
  1
  2   5
  3   8  14
  4  11  20  30
  5  14  26  40  55
  6  17  32  50  70  91
  7  20  38  60  85 112 140
  ...

Robert Israel, Table of n, a(n) for n = 1..10011
Antonio Bernini, Matteo Cervetti, Luca Ferrari, and Einar Steingrimsson, Enumerative combinatorics of intervals in the Dyck pattern poset, arXiv:1910.00299 [math.CO], 2019. See p. 5.

Cf. A083003, A083487. Right side of triangle gives A000330.
Main diagonal is A000330, row sums are A001296. - Paul D. Hanna and other correspondents, May 28 2003
Cf. A130684. - Joel B. Lewis

/* As triangle */ [[(k+3*k*n+3*k^2*n-k^3)/6: k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Mar 26 2019

f:=proc(m,n) add((m-i)*(n-i),i=0..min(m,n)); end;

T[n_, k_] := Sum[(n-i)(k-i), {i, 0, Min[n, k]}];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2019 *)

T(n, k) = ( k + 3*k*n + 3*k^2*n - k^3 ) / 6.
T(n, k) = Sum_{i=0..min(n,k)} (n-i)*(k-i). - N. J. A. Sloane, Nov 17 2007
G.f.: (1+x*y-2*x^2*y)*x*y/((1-x*y)^4*(1-x)^2). - Robert Israel, Dec 20 2017

Edited by Robert Israel, Dec 20 2017

A105552 Irregular triangle T(n,k) read down columns: the number of compositions c of n with largest_part(c)+length(c)=k+1 in row n, column k.

Original entry on oeis.org

1, 2, 4, 1, 7, 5, 2, 11, 14, 12, 5, 1, 16, 30, 39, 32, 18, 7, 2, 22, 55, 95, 113, 101, 71, 41, 18, 6, 1, 29, 91, 195, 299, 357, 350, 292, 207, 126, 64, 27, 9, 2, 37, 140, 357, 664, 978, 1204, 1283, 1198, 992, 731, 482, 284, 148, 66, 25, 7, 1, 46, 204, 602, 1309, 2274, 3329, 4253

Offset: 1

Alford Arnold, May 03 2005

For each of the A000041(n) partitions of n, one can assign a weight to the partition which counts the permutations of that partition, given by the multinomial coefficient derived from the frequency representation of the parts.
An equivalent representation is given by writing down all compositions of n.
The entries count those partitions multiplied by their weights (=compositions) of n where the sum of the largest addend plus number of parts equals k+1. Only nonzero counts are entered into the sequence.
Each entry can also be interpreted as counting a subset of numbers in A055932, because there is a 1-to-1 correspondence between their prime signature and ordered partitions.
Each diagonal of T(n,k) can be decomposed into p(n) sequences. For example,
A086602 = 2 12 39 95 195 ... is the sum of
A000330 = 1 5 14 30 55 ... plus
A001296 = 1 7 25 65 140 ...
The main diagonal and subdiagonals in order of appearance are A000124, A000330, A086602, A089574, A107600, A107601, A109125, ...

			The row n=7 starts from the partitions (weights in parentheses) 7 (1), 6+1 (2), 5+2 (2), 4+3 (2), 5+1+1 (3), 4+2+1 (6=3!/1!/1!/1!), 3+3+1 (3), 3+2+2 (3), 4+1+1+1 (4=4!/1!/3!), 3+2+1+1 (12 = 4!/1!/1!/2!), 2+2+2+1 (4), 3+1+1+1+1+1 (5), 2+2+1+1+1 (10=5!/2!/3!), 2+1+1+1+1 (6), 1+1+1+1+1+1 (1).
Then T(7,7) = 1+2+3+4+5+6+1 = 22 is the sum of the weights of partitions with largest part 7 and length 1, largest part 6 and length 2,... largest part 1 and length 7.
T(7,6) = 2+6+12+10 = 30 is the sum of the weights of the partitions with largest part 6 and length 1, largest part 5 and length 2, ..., largest part 1 and length 6.
T(7,5) = 2+3+3+4 = 12 collects all the partitions with largest part 5 and length 1 down to largest part 1 and length 5.
The array has A033638(k) nonzero entries per column, starting at n=1 as :
1
..2
....4
....1..7
.......5..11
.......2..14..16
..........12..30..22
...........5..39..55..29
...........1..32..95..91..37
..............18.113.195.140
...............7.101.299.357
...............2

Alois P. Heinz, Columns k = 1..40, flattened

Cf. A047969, A047970, A055932, A057335, A083480, A083906, A089349, A033638, A086602 (subdiagonal), A089574 (subdiagonal).

A033638 := proc(n) ( (7+(-1)^n)/2 + n^2 )/4 ; end proc:
freq := proc(L,n) local a,p; a := 0 ; for p in L do if p = n then a := a+1 ; end if; end do: a ; end proc:
M3 := proc(L) local a,i; a := factorial(nops(L)) ; for i in convert(L,set) do a := a/factorial(freq(L,i)) ; end do: a ; end proc:
A105552 := proc(n,k) local p,a,l ; a := 0 ; for p in combinat[partition](n) do if max(op(p)) + nops(p) = k+1 then a := a+ M3(p); end if; end do ; a ; end proc:
for k from 1 to 15 do for n from k to k+A033638(k)+1 do T := A105552(n,k) ; if T >0 then printf("%d,", A105552(n,k)) ; end if; end do: printf("\n") ; end do: # R. J. Mathar, Jun 26 2010
# second Maple program:
b:= proc(n, k, p) option remember; `if`(n=0 and k=0, 1,
     `if`(k<1, 0, add(b(n-j, k-1-max(p, j)+p, max(p, j)), j=1..n)))
    end:
T:= k-> seq(b(n, k+1, 0), n=k..k+floor((k-1)^2/4)):
seq(T(k), k=1..10);  # Alois P. Heinz, Jul 24 2013

b[n_, k_, p_] := b[n, k, p] = If[n == 0 && k == 0, 1, If[k < 1, 0, Sum[b[n - j, k - 1 - Max[p, j] + p, Max[p, j]], {j, 1, n}]]]; T[k_] := Table[b[n, k + 1, 0], {n, k, k + Floor[(k - 1)^2/4]}]; Table[T[k], {k, 1, 10}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

Row sums: Sum_{k=0..n} T(n,k) = 2^(n-1).

Column sums: Sum_{n>=k} T(n,k) = A047970(n).

Definition clarified by R. J. Mathar, Jun 26 2010

A125232 Triangle T(n,k) read by rows: the (n-k)-th term of the k-fold iterated partial sum of the pentagonal numbers.

Original entry on oeis.org

1, 5, 1, 12, 6, 1, 22, 18, 7, 1, 35, 40, 25, 8, 1, 51, 75, 65, 33, 9, 1, 70, 126, 140, 98, 42, 10, 1, 92, 196, 266, 238, 140, 52, 11, 1, 117, 288, 462, 504, 378, 192, 63, 12, 1, 145, 405, 750, 966, 882, 570, 255, 75, 13, 1, 176, 550, 1155, 1716, 1848, 1452, 825, 330, 88, 14, 1

Offset: 1

Gary W. Adamson, Nov 24 2006

			First few rows of the triangle are:
   1;
   5,   1;
  12,   6,   1;
  22,  18,   7,   1;
  35,  40,  25,   8,   1;
  51,  75,  65,  33,   9,   1;
  70, 126, 140,  98,  42,  10,   1;
  ...
Example: (5,3) = 65 = 25 + 40 = (4,3) + (4,2).

Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966, p 189.

Robert Israel, Table of n, a(n) for n = 1..10011(rows 0 to 140, flattened)

Columns: A000326 (pentagonal numbers), A002411, A001296, A051836, A051923.

Cf. A095264 (row sums).

A125232 := proc(n,k) option remember ; if k = 0 then A000326(n) ; elif k = n-1 then 1 ; else procname(n-1,k)+procname(n-1,k-1) ; fi : end: # R. J. Mathar, Jun 09 2008

nmax = 11; col[1] = Table[n(3n-1)/2, {n, 1, nmax}]; col[k_] := col[k] = Prepend[Accumulate[col[k-1]], 0]; Table[col[k][[n]], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2019 *)

T(n,0)=A000326(n). T(n,k)=T(n-1,k) + T(n-1,k-1), k>0. - R. J. Mathar, Jun 09 2008

G.f. as triangle: (1+2*x)/((1-x)^2*(1-x-x*y)). - Robert Israel, Nov 07 2016

Edited and extended by R. J. Mathar, Jun 09 2008

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

2, 9, 5, 24, 18, 8, 50, 42, 27, 11, 90, 80, 60, 36, 14, 147, 135, 110, 78, 45, 17, 224, 210, 180, 140, 96, 54, 20, 324, 308, 273, 225, 170, 114, 63, 23, 450, 432, 392, 336, 270, 200, 132, 72, 26, 605, 585, 540, 476, 399, 315

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213820.
Antidiagonal sums: A153978.
Row 1, (1,2,3,4,...)**(2,5,8,11,...): A006002.
Row 2, (1,2,3,4,...)**(5,8,11,14,...): is it the sequence A212343?.
Row 3, (1,2,3,4,...)**(8,11,14,17,...): (k^3 + 8*k^2 + 7*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
2....9....24....50....90....147
5....18...42....80....135...210
8....27...60....110...180...273
11...36...78....140...225...336
14...45...96....170...270...399
17...54...114...200...315...462

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A212500

b[n_]:=n;c[n_]:=3n-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}] (* A213819 *)
Table[t[n,n],{n,1,40}] (* A213820 *)
d/2 (* A002414 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A153978 *)
s1/2 (* A001296 *)

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) = x(3*n-1 - (3*n-4)*x) and g(x) = (1-x)^4.

A242322 T(n,k)=Number of length n+k+2 0..k arrays with every value 0..k appearing at least once in every consecutive k+3 elements, and new values 0..k introduced in order.

Original entry on oeis.org

7, 25, 13, 65, 61, 24, 140, 185, 145, 44, 266, 440, 503, 337, 81, 462, 896, 1300, 1316, 781, 149, 750, 1638, 2801, 3648, 3398, 1829, 274, 1155, 2766, 5334, 8231, 10012, 8801, 4269, 504, 1705, 4395, 9290, 16194, 23486, 27368, 23069, 9957, 927, 2431, 6655

Offset: 1

R. H. Hardin, May 10 2014

Table starts
....7....25.....65.....140.....266.....462......750.....1155.....1705.....2431
...13....61....185.....440.....896....1638.....2766.....4395.....6655.....9691
...24...145....503....1300....2801....5334.....9290....15123....23350....34551
...44...337...1316....3648....8231...16194....28897....47931....75118...112511
...81...781...3398...10012...23486...47466....86381...145443...230647...348771
..149..1829...8801...27368...66366..137166...253674...432331...692113..1054531
..274..4269..23069...75236..187671..395166...740496..1274419..2055676..3150991
..504..9957..60197..208976..533801.1141290..2161503..3749211..6083896..9369751
..927.23233.156887..577964.1530356.3312546..6326951.11042115.18002245.27827211
.1705.54225.408962.1596216.4371836.9669270.18590776.32600811.53341987.82686971

			Some solutions for n=5 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....0....1....1....1....1....1....1....1....1....0....1....0....1....1
..0....2....1....1....2....2....1....2....2....2....1....0....2....1....0....1
..2....1....2....2....3....2....0....1....3....3....2....1....3....2....2....2
..3....3....3....3....0....3....2....3....1....4....3....2....4....0....0....0
..4....0....0....0....2....2....3....4....0....2....2....3....3....3....3....3
..1....4....4....4....4....4....4....0....4....0....4....4....2....4....4....4
..2....1....2....4....1....0....1....2....3....3....0....1....0....1....4....4
..0....1....0....1....1....1....1....3....3....1....1....2....1....1....1....1
..3....2....1....2....3....4....0....1....2....1....4....0....3....3....2....2
..1....0....4....0....2....4....0....1....2....2....4....4....2....2....0....4

R. H. Hardin, Table of n, a(n) for n = 1..369

Column 1 is A000073(n+5)

Row 1 is A001296(n+1)

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2) +a(n-3)
k=2: a(n) = a(n-1) +2*a(n-2) +2*a(n-3) +2*a(n-4) -a(n-5) -a(n-6)
k=3: [order 10]
k=4: [order 15]
k=5: [order 21]
k=6: [order 28]
Empirical for row n:
n=1: a(n) = (1/8)*n^4 + (11/12)*n^3 + (19/8)*n^2 + (31/12)*n + 1
n=2: a(n) = (5/8)*n^4 + (35/12)*n^3 + (39/8)*n^2 + (43/12)*n + 1
n=3: a(n) = (21/8)*n^4 + (89/12)*n^3 + (67/8)*n^2 + (55/12)*n + 1
n=4: a(n) = (77/8)*n^4 + (179/12)*n^3 + (103/8)*n^2 + (67/12)*n + 1
n=5: a(n) = (261/8)*n^4 + (245/12)*n^3 + (163/8)*n^2 + (79/12)*n + 1
n=6: a(n) = (845/8)*n^4 - (73/12)*n^3 + (343/8)*n^2 + (91/12)*n + 1 for n>1
n=7: a(n) = (2661/8)*n^4 - (2263/12)*n^3 + (1059/8)*n^2 + (103/12)*n + 1 for n>2
n=8: a(n) = (8237/8)*n^4 - (11701/12)*n^3 + (3879/8)*n^2 + (115/12)*n + 1 for n>3
n=9: a(n) = (25221/8)*n^4 - (46531/12)*n^3 + (14275/8)*n^2 + (127/12)*n + 1 for n>4
n=10: a(n) = (76685/8)*n^4 - (165601/12)*n^3 + (50455/8)*n^2 + (139/12)*n + 1 for n>5
n=11: a(n) = (232101/8)*n^4 - (555055/12)*n^3 + (171139/8)*n^2 + (151/12)*n + 1 for n>6
n=12: a(n) = (700397/8)*n^4 - (1794061/12)*n^3 + (561703/8)*n^2 + (163/12)*n + 1 for n>7
n=13: a(n) = (2109381/8)*n^4 - (5664667/12)*n^3 + (1798755/8)*n^2 + (175/12)*n + 1 for n>8
n=14: a(n) = (6344525/8)*n^4 - (17608249/12)*n^3 + (5657175/8)*n^2 + (187/12)*n + 1 for n>9
n=15: a(n) = (19066341/8)*n^4 - (54151687/12)*n^3 + (17559907/8)*n^2 + (199/12)*n + 1 for n>10

cp's OEIS Frontend

A254142 a(n) = (9*n+10)*binomial(n+9,9)/10.

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A005460 a(n) = (3*n+4)*(n+3)!/24.

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A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

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A213086 Square array read by antidiagonals: T(n,m) (n>=1, m>=0) is the number of partitions of mn that are the sum of m not necessarily distinct partitions of n.

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A037960 a(n) = n*(3*n+1)*(n+2)!/24.

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A082652 Triangle read by rows: T(n,k) is the number of squares that can be found in a k X n rectangular grid of little squares, for 1 <= k <= n.

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A254142 a(n) = (9n+10)binomial(n+9,9)/10.

A005460 a(n) = (3n+4)(n+3)!/24.

A037960 a(n) = n(3n+1)*(n+2)!/24.

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.