A077028 -id:A077028 - cp's OEIS frontend

A309559 Triangle read by rows: T(n,k) = 1 + n + k^2/2 - k/2 + k*(n-k), n >= 0, 0 <= k <= n.

1, 2, 2, 3, 4, 4, 4, 6, 7, 7, 5, 8, 10, 11, 11, 6, 10, 13, 15, 16, 16, 7, 12, 16, 19, 21, 22, 22, 8, 14, 19, 23, 26, 28, 29, 29, 9, 16, 22, 27, 31, 34, 36, 37, 37, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 11, 20, 28, 35, 41, 46, 50, 53, 55, 56, 56, 12, 22, 31, 39, 46, 52, 57, 61, 64, 66, 67, 67, 13, 24, 34, 43, 51, 58, 64, 69, 73, 76, 78, 79, 79

Offset: 0

Philip K Hotchkiss, Aug 07 2019

The rascal triangle (A077028) can be generated by the rule South = (East*West+1)/North or South = East+West+1-North; this number triangle can also be generated by South = East+West+1-North, but there not by an equation of the form South = (East*West+d)/North.

			For row n=3: T(3,0)=4, T(3,1)=6, T(3,2)=6, T(3,3)=7.
Triangle T begins:
                  1
                2   2
              3   4   4
            4   6   7   7
          5   8  10  11  11
        6  10  13  15  16  16
      7  12  16  19  21  22  22
    8  14  19  23  26  28  29  29
  9  16  22  27  31  34  36  37  37
                 ...

Philip K Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019.

Cf. A077028, A309555, A309557.

T := proc(n, k)
   if n<0 or k<0 or k>n then
       0;
   else
       1+n+(1/2)*k^2-(1/2)k +k*(n-k);
   end if;

T[n_,k_]:=1+n+(1/2)*k^2-(1/2)k +k*(n-k); Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
f[n_] := Table[SeriesCoefficient[(-1+(3-2*x)*y+(-1+x)*y^2)/((-1+x)^2*(-1+y)^3), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 13,0]] (* Stefano Spezia, Sep 08 2019 *)

G.f.: (-1+(3-2*x)*y+(-1+x)*y^2)/((-1+x)^2*(-1+y)^3). - Stefano Spezia, Sep 08 2019

A053730 a(n) = 2^(n-2)*(n^2 - n + 4).

Original entry on oeis.org

1, 2, 6, 20, 64, 192, 544, 1472, 3840, 9728, 24064, 58368, 139264, 327680, 761856, 1753088, 3997696, 9043968, 20316160, 45350912, 100663296, 222298112, 488636416, 1069547520, 2332033024, 5066719232, 10972299264, 23689428992

Offset: 0

N. J. A. Sloane, Mar 24 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A053545.

List([0..30], n-> 2^(n-2)*(n^2 -n +4)); # G. C. Greubel, Sep 06 2019

I:=[1, 2, 6]; [n le 3 select I[n] else 6*Self(n-1)-12*Self(n-2) +8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 28 2012

seq(2^(n-2)*(n^2 -n +4), n=0..30); # G. C. Greubel, Sep 06 2019

CoefficientList[Series[(1-4*x+6*x^2)/(1-2*x)^3,{x,0,30}],x] (* Vincenzo Librandi, Apr 28 2012 *)
LinearRecurrence[{6,-12,8}, {1,2,6}, 30] (* G. C. Greubel, Sep 06 2019 *)

vector(30, n, 2^(n-3)*(n^2 -3*n +6)) \\ G. C. Greubel, Sep 06 2019

[2^(n-2)*(n^2 -n +4) for n in (0..30)] # G. C. Greubel, Sep 06 2019

G.f.: (1-4*x+6*x^2)/(1-2*x)^3. - Colin Barker, Apr 01 2012
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3). - Vincenzo Librandi, Apr 28 2012
a(n) = Sum_{k=0..n} binomial(n,k) * A077028(n,k), where A077028(n,k) = (n-k)*k + 1. - Paul D. Hanna, Oct 11 2015

A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 13, 7, 1, 1, 15, 25, 22, 9, 1, 1, 21, 41, 46, 33, 11, 1, 1, 28, 61, 79, 73, 46, 13, 1, 1, 36, 85, 121, 129, 106, 61, 15, 1, 1, 45, 113, 172, 201, 191, 145, 78, 17, 1, 1, 55, 145, 232, 289, 301, 265, 190, 97, 19, 1

Offset: 0

Peter Luschny, Mar 21 2023

A combination of a multiset M is an unordered selection of k objects of M, where every object can appear at most as many times as it appears in M.
A(n, k) = Cardinality(Union_{j=0..k} Combination(MultiSet(1^[j*n], 0^[(k-j)*n]))), where MultiSet(r^[s], u^[v]) denotes a set that contains the element r with multiplicity s and the element u with multiplicity v; thus the multisets under consideration have n*k elements. Since the base set is {1, 0} the elements can be represented as binary strings. Applying the combination operator to the multisets results in a set of binary strings where '0' resp. '1' can appear at most j*n resp. (k-j)*n times. 'At most' means that they do not have to appear; in other words, the resulting set always includes the empty string ''.
In contrast to the procedure in A361045 we consider here the cardinality of the set union and not the sum of the individual cardinalities. If you want to exclude the empty string, you will find the sequences listed in A361521. The same construction with multiset permutations instead of multiset combinations results in A361043.
A different view can be taken if one considers the hypergeometric representation, hypergeom([-k, -m], [1], n). This is a family of arrays that includes the 'rascal' triangle: the all 1's array A000012 (m = 0), the rascal array A077028 (m = 1), this array (m = 2), and A361731 (m = 3).

			Array A(n, k) starts:
   [0] 1,  1,   1,    1,   1,   1,   1,    1, ...  A000012
   [1] 1,  3,   6,   10,  15,  21,  28,   36, ...  A000217
   [2] 1,  5,  13,   25,  41,  61,  85,  113, ...  A001844
   [3] 1,  7,  22,   46,  79, 121, 172,  232, ...  A038764
   [4] 1,  9,  33,   73, 129, 201, 289,  393, ...  A081585
   [5] 1, 11,  46,  106, 191, 301, 436,  596, ...  A081587
   [6] 1, 13,  61,  145, 265, 421, 613,  841, ...  A081589
   [7] 1, 15,  78,  190, 351, 561, 820, 1128, ...  A081591
   000012  | A028872 | A239325 |
       A005408    A100536   A069133
.
Triangle T(n, k) starts:
   [0] 1;
   [1] 1,  1;
   [2] 1,  3,   1;
   [3] 1,  6,   5,   1;
   [4] 1, 10,  13,   7,   1;
   [5] 1, 15,  25,  22,   9,   1;
   [6] 1, 21,  41,  46,  33,  11,   1;
   [7] 1, 28,  61,  79,  73,  46,  13,  1;
   [8] 1, 36,  85, 121, 129, 106,  61, 15,  1;
   [9] 1, 45, 113, 172, 201, 191, 145, 78, 17, 1.
.
Row 4 of the triangle:
A(0, 4) =  1 = card('').
A(1, 3) = 10 = card('', 0, 00, 000, 1, 10, 100, 11, 110, 111).
A(2, 2) = 13 = card('', 0, 00, 000, 0000, 1, 10, 100, 11, 110, 1100, 111, 1111).
A(3, 1) =  7 = card('', 0, 00, 000, 1, 11, 111).
A(4, 0) =  1 = card('').

Rows: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591.
Columns: A000012, A005408, A028872, A100536, A239325, A069133.
Cf. A239592 (main diagonal), A239331 (transposed array).
Cf. A361045, A361043, A361521, A361731, A077028.

A := (n, k) -> 1 + n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;
# Alternative:
ogf := n -> (1 + (n - 1)*x)^2 / (1 - x)^3:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..7);

def A(m: int, steps: int) -> int:
    if m == 0: return 1
    size = m * steps
    cset = set()
    for a in range(0, size + 1, m):
        S = [str(int(i < a)) for i in range(size)]
        C = Combinations(S)
        cset.update("".join(i for i in c) for c in C)
    return len(cset)
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size + 1)]
for n in range(8): print(ARow(n, 7))

A(n, k) = 1 + n*k*(4 + n*(k - 1))/2.
T(n, k) = 1 + k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = [x^k] (1 + (n - 1)*x)^2 / (1 - x)^3.
A(n, k) = hypergeom([-k, -2], [1], n).
A(n, k) = A361521(n, k) + 1.

A374378 Iterated rascal triangle R2: T(n,k) = Sum_{m=0..2} binomial(n-k,m)*binomial(k,m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 19, 15, 6, 1, 1, 7, 21, 31, 31, 21, 7, 1, 1, 8, 28, 46, 53, 46, 28, 8, 1, 1, 9, 36, 64, 81, 81, 64, 36, 9, 1, 1, 10, 45, 85, 115, 126, 115, 85, 45, 10, 1, 1, 11, 55, 109, 155, 181, 181, 155, 109, 55, 11, 1

Offset: 0

Kolosov Petro, Jul 06 2024

Triangle T(n,k) is the second triangle R2 among the rascal-family triangles; A374452 is triangle R3; A077028 is triangle R1.
Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 5).
Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 2).

			Triangle begins:
--------------------------------------------------
k=     0   1   2   3    4    5    6   7   8   9 10
--------------------------------------------------
n=0:   1
n=1:   1   1
n=2:   1   2   1
n=3:   1   3   3   1
n=4:   1   4   6   4    1
n=5:   1   5  10  10    5    1
n=6:   1   6  15  19   15    6    1
n=7:   1   7  21  31   31   21    7   1
n=8:   1   8  28  46   53   46   28   8   1
n=9:   1   9  36  64   81   81   64  36   9   1
n=10:  1  10  45  85  115  126  115  85  45  10  1

Kolosov Petro, Table of n, a(n) for n = 0..1325
Amelia Gibbs and Brian K. Miceli, Two Combinatorial Interpretations of Rascal Numbers, arXiv:2405.11045 [math.CO], 2024.
Jena Gregory, Brandt Kronholm, and Jacob White, Iterated rascal triangles, Aequationes mathematicae, 2023.
Jena Gregory, Iterated rascal triangles, Theses and Dissertations. 1050., The University of Texas Rio Grande Valley, 2022.
Philip K. Hotchkiss, Student Inquiry and the Rascal Triangle, arXiv:1907.07749 [math.HO], 2019.
Philip K. Hotchkiss, Generalized Rascal Triangles, Journal of Integer Sequences, Vol. 23, 2020.
Petro Kolosov, Identities in Iterated Rascal Triangles, 2024.

Cf. A077028, A007318, A000027, A000217, A005448, A056108, A212656, A000292, A095667, A095666, A006261.

t[n_, k_]:=Sum[Binomial[n - k, m]*Binomial[k, m], {m, 0, 2}]; Column[Table[t[n, k], {n, 0, 12}, {k, 0, n}], Center]

T(n,k) = 1 + k*(n-k) + (1/4)*(k-1)*k*(n-k-1)*(n-k).
Row sums give A006261(n).
Diagonal T(n+1, n) gives A000027(n).
Diagonal T(n+2, n) gives A000217(n).
Diagonal T(n+3, n) gives A005448(n).
Diagonal T(n+4, n) gives A056108(n).
Diagonal T(n+5, n) gives A212656(n).
Column k=3 difference binomial(n+6, 3) - T(n+6, 3) gives C(n+3,3)=A007318(n+3,3).
Column k=4 difference binomial(n+7, 4) - T(n+7, 4) gives fifth column of (1,4)-Pascal triangle A095667.
G.f.: (1 + 3*x^4*y^2 - (2*x + 3*x^3*y)*(1 + y) + x^2*(1 + 5*y + y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 09 2024

A105851 Binomial transform triangle, read by rows.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 20, 12, 5, 1, 32, 48, 32, 16, 6, 1, 64, 112, 80, 44, 20, 7, 1, 128, 256, 192, 112, 56, 24, 8, 1, 256, 576, 448, 272, 144, 68, 28, 9, 1, 512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1, 1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1

Offset: 0

Gary W. Adamson, Apr 23 2005

Let P = Pascal's triangle as an infinite lower triangular matrix and A is the infinite array of arithmetic sequences as shown in A077028:
  1, 1, 1,  1,  1, ...
  1, 2, 3,  4,  5, ...
  1, 3, 5,  7,  9, ...
  1, 4, 7, 10, 13, ...
  1, 5, 9, 13, 17, ...
Perform the operation P * A, getting a new array with each column being the binomial transform of an arithmetic sequence. Take antidiagonals of the new array, then by rows = the triangle of A105851.

			Column 3: 1, 5, 16, 44, 112, ... (A053220) is the binomial transform of 3k+1 (A016777: 1, 4, 7, ...).
Triangle begins:
     1;
     2,    1;
     4,    3,    1;
     8,    8,    4,    1;
    16,   20,   12,    5,   1;
    32,   48,   32,   16,   6,   1;
    64,  112,   80,   44,  20,   7,   1;
   128,  256,  192,  112,  56,  24,   8,  1;
   256,  576,  448,  272, 144,  68,  28,  9,  1;
   512, 1280, 1024,  640, 352, 176,  80, 32, 10,  1;
  1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1;
  ...

Cf. A077028, A001792, A001787, A053220, A016777, A014480.

/* As triangle */ [[(2+k*(n-k))*2^(n-k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 26 2015

seq(seq((2 + k*(n - k))*2^(n-k-1),k=0..n),n=0..10); # Peter Bala, Jul 26 2015

t[n_, k_]:=(2 + k (n - k)) 2^(n - k - 1); Table[t[n - 1, k - 1], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jul 26 2015 *)

n-th column of the triangle is the binomial transform of the arithmetic sequence (n*k + 1), (k = 0, 1, 2, ...).
From Peter Bala, Jul 26 2015: (Start)
T(n,k) = (2 + k*(n - k))*2^(n-k-1) for 0 <= k <= n.
O.g.f.: (1 - x*(2 + t) + 3*t*x^2)/((1 - 2*x)^2*(1 - t*x)^2) = 1 + (2 + t)*x + (4 + 3*t + t^2)*x^2 + ....
k-th column g.f.: (1 + (k - 2)*x)/(1 - 2*x)^2. Cf. A077028. (End)

More terms from Philippe Deléham, Mar 31 2007

A128139 Triangle read by rows: matrix product A004736 * A128132.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 7, 5, 1, 6, 9, 10, 9, 6, 1, 7, 11, 13, 13, 11, 7, 1, 8, 13, 16, 17, 16, 13, 8, 1, 9, 15, 19, 21, 21, 19, 15, 9, 1, 10, 17, 22, 25, 26, 25, 22, 17, 10

Offset: 0

Gary W. Adamson, Feb 16 2007

A077028 with the final term in each row omitted.
Interchanging the factors in the matrix product leads to A128140 = A128132 * A004736.
From Gary W. Adamson, Jul 01 2012: (Start)
Alternatively, antidiagonals of an array A(n,k) of sequences with arithmetic progressions as follows:
  1, 2, 3,  4,  5,  6, ...
  1, 3, 5,  7,  9, 11, ...
  1, 4, 7, 10, 13, 16, ...
  1, 5, 9, 13, 17, 21, ...
  ... (End)
From Gary W. Adamson, Jul 02 2012: (Start)
A summation generalization for Sum_{k>=1} 1/(A(n,k)*A(n,k+1)) (formulas copied from A002378, A000466, A085001, A003185):
  1 = 1/(1)*(2) + 1/(2)*(3) + 1/(3)*(4)  + ...
  1 = 2/(1)*(3) + 2/(3)*(5) + 2/(5)*(7)  + ...
  1 = 3/(1)*(4) + 3/(4)*(7) + 3/(7)*(10) + ...
  1 = 4/(1)*(5) + 4/(5)*(9) + 4/(9)*(13) + ...
  ...
As a summation of terms equating to a definite integral:
  Integral_{0..1} dx/(1+x) = ... 1 - 1/2 + 1/3 - 1/4 + ... = log(2).
  Integral_{0..1} dx/(1+x^2) = 1 - 1/3 + 1/5 - 1/7  + ... = Pi/4 (see A157142)
  Integral_{0..1} dx/(1+x^3) = 1 - 1/4 + 1/7 - 1/10 + ... (see A016777)
  Integral_{0..1} dx/(1+x^4) = 1 - 1/5 + 1/9 - 1/13 + ... (see A016813). (End)

			First few rows of the triangle:
  1;
  1,  2;
  1,  3,  3;
  1,  4,  5,  4;
  1,  5,  7,  7,  5;
  1,  6,  9, 10,  9,  6;
  1,  7, 11, 13, 13, 11,  7;
  1,  8, 13, 16, 17, 16, 13,  8;
  1,  9, 15, 19, 21, 21, 19, 15,  9;
  1, 10, 17, 22, 25, 26, 25, 22, 17, 10;
  ...

Cf. A004736, A128132, A128140, A004006 (row sums).

A004736 * A128132 as infinite lower triangular matrices.

T(n,k) = k*(1+n-k)+1 = 1 + A094053(n+1,1+n-k). - R. J. Mathar, Jul 09 2012

A134398 Triangle read by rows: T(n, k) = (k-1)*(n-k) + binomial(n-1,k-1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 10, 7, 1, 1, 9, 16, 16, 9, 1, 1, 11, 23, 29, 23, 11, 1, 1, 13, 31, 47, 47, 31, 13, 1, 1, 15, 40, 71, 86, 71, 40, 15, 1, 1, 17, 50, 102, 146, 146, 102, 50, 17, 1, 1, 19, 61, 141, 234, 277, 234, 141, 61, 19, 1

Offset: 1

Gary W. Adamson, Oct 23 2007

Row sums = A116725: (1, 2, 5, 12, 26, 52, ...).

			First few rows of the triangle are:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  5,  1;
  1,  7, 10,  7,  1;
  1,  9, 16, 16,  9,  1;
  1, 11, 23, 29, 23, 11,  1;
  1, 13, 31, 47, 47, 31, 13,  1;
  1, 15, 40, 71, 86, 71, 40, 15, 1;
...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A007318, A077028, A116725.

Flat(List([1..10], n-> List([1..n], k-> (k-1)*(n-k) + Binomial(n-1,k-1) ))); # G. C. Greubel, Nov 29 2019

[(k-1)*(n-k) + Binomial(n-1,k-1): k in [1..n], n in [1..10]]; // G. C. Greubel, Nov 29 2019

seq(seq( (k-1)*(n-k) + binomial(n-1,k-1), k=1..n), n=1..10); # G. C. Greubel, Nov 29 2019

p[x_, n_]:= p[x,n]= If[n==0, 1, (x+1)^n +Sum[(n-m)*m*x^m*(1 +x^(n-2*m)), {m, 1, n- 1}]/2]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]//Flatten (* Roger L. Bagula, Nov 02 2008 *)
Table[(k-1)*(n-k) + Binomial[n-1, k-1], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Nov 29 2019 *)

T(n,k) = (k-1)*(n-k) + binomial(n-1,k-1); \\ G. C. Greubel, Nov 29 2019

[[(k-1)*(n-k) + binomial(n-1,k-1) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Nov 29 2019

T(n, k) = A077028(n,k) + A007318(n,k) - 1.
Let p(x, n) = (1+x)^n + (1/2) * Sum_{j=1..n-1} (n-j)*j*x^j*(1 + x^(n - 2*j)) with p(x, 0) = 1, then T(n, k) = Coefficients(p(x,n)). - Roger L. Bagula, Nov 02 2008
T(n, k) = (k-1)*(n-k) + binomial(n-1,k-1). - G. C. Greubel, Nov 29 2019

Extended by Roger L. Bagula, Nov 02 2008

Edited by G. C. Greubel, Nov 29 2019

A176270 Triangle T(n,m) = 1 + m*(m-n) read by rows, 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -3, -2, 1, 1, -3, -5, -5, -3, 1, 1, -4, -7, -8, -7, -4, 1, 1, -5, -9, -11, -11, -9, -5, 1, 1, -6, -11, -14, -15, -14, -11, -6, 1, 1, -7, -13, -17, -19, -19, -17, -13, -7, 1, 1, -8, -15, -20, -23, -24, -23, -20, -15, -8, 1

Offset: 0

Roger L. Bagula, Apr 13 2010

For GCD(-1 - m,-1 - n + m) = 1, smallest number that cannot be written as a*(-1 - m) + b*(-1 - n + m) with a and b in the nonnegative integers. - Thomas Anton, May 22 2019

			Triangle begins
  1;
  1,   1;
  1,   0,   1;
  1,  -1,  -1,   1;
  1,  -2,  -3,  -2,   1;
  1,  -3,  -5,  -5,  -3,   1;
  1,  -4,  -7,  -8,  -7,  -4,   1;
  1,  -5,  -9, -11, -11,  -9,  -5,   1;
  1,  -6, -11, -14, -15, -14, -11,  -6,   1;
  1,  -7, -13, -17, -19, -19, -17, -13,  -7,   1;
  1,  -8, -15, -20, -23, -24, -23, -20, -15,  -8,   1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A005586 (row sums), A077028.

Flat(List([0..12], n-> List([0..n], k-> k*(k-n)+1 ))); # G. C. Greubel, May 30 2019

[[k*(k-n)+1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 30 2019

A176270 := proc(n,m)
        1+m*(m-n) ;
end proc: # R. J. Mathar, May 03 2013

Table[k*(k-n)+1, {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 30 2019 *)

{T(n,k) = k*(k-n)+1}; \\ G. C. Greubel, May 30 2019

[[k*(k-n)+1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 30 2019

T(n,m) = binomial(n-m+1,2) + binomial(m+1,2) - binomial(n+1,2) + 1 = m^2 - n*m + 1.
T(n,m) = T(n,n-m).
T(n,m) = 2 - A077028(n,m) for 0 <= m <= n. - Werner Schulte, Nov 10 2020

Edited by R. J. Mathar, May 03 2013

A176282 Triangle T(n,k) = 1 + A000330(n) - A000330(k) - A000330(n-k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 21, 16, 1, 1, 25, 37, 37, 25, 1, 1, 36, 57, 64, 57, 36, 1, 1, 49, 81, 97, 97, 81, 49, 1, 1, 64, 109, 136, 145, 136, 109, 64, 1, 1, 81, 141, 181, 201, 201, 181, 141, 81, 1, 1, 100, 177, 232, 265, 276, 265, 232, 177, 100, 1

Offset: 0

Roger L. Bagula, Apr 14 2010

Not summing squares but summing integers implied by the definition (i.e., not using A000330 but A000217) gives A077028.

Row sums = {1, 2, 6, 20, 55, 126, 252, 456, 765, 1210, 1826, ...} = (n+1)*(n+2)*(n^2-2*n+3)/6.

			Triangle begins as:
  1;
  1,   1;
  1,   4,   1;
  1,   9,   9,   1;
  1,  16,  21,  16,   1;
  1,  25,  37,  37,  25,   1;
  1,  36,  57,  64,  57,  36,   1;
  1,  49,  81,  97,  97,  81,  49,   1;
  1,  64, 109, 136, 145, 136, 109,  64,   1;
  1,  81, 141, 181, 201, 201, 181, 141,  81,   1;
  1, 100, 177, 232, 265, 276, 265, 232, 177, 100, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A077028.

Flat(List([0..12], n-> List([0..n], k-> 1 + k*(n+1)*(n-k) ))); # G. C. Greubel, Nov 24 2019

[1 + k*(n+1)*(n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 24 2019

seq(seq(1 + k*(n+1)*(n-k), k=0..n), n=0..12); # G. C. Greubel, Nov 24 2019

(* Set of sequences q=1..10. This sequence is q=2. *)
f[n_, k_, q_]:= f[n, k, q] = 1 + Sum[i^q, {i,0,n}] - Sum[i^q, {i,0,k}] - Sum[i^q, {i,0,n-k}]; Table[Flatten[Table[f[n, k, q], {n,0,12}, {k,0,n}]], {q,1,10}]
(* Second program *)
Table[1 + k*(n+1)*(n-k), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 24 2019 *)

T(n,k) = 1 + k*(n+1)*(n-k); \\ G. C. Greubel, Nov 24 2019

[[1 + k*(n+1)*(n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 24 2019

T(n,k) = T(n,n-k).

T(n,k) = 1 + k*(n+1)*(n-k). - G. C. Greubel, Nov 24 2019

Edited by R. J. Mathar, May 03 2013

A275490 Square array of 5D pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 5, 1, 6, 15, 1, 7, 21, 35, 1, 8, 27, 56, 70, 1, 9, 33, 77, 126, 126, 1, 10, 39, 98, 182, 252, 210, 1, 11, 45, 119, 238, 378, 462, 330, 1, 12, 51, 140, 294, 504, 714, 792, 495, 1, 13, 57, 161, 350, 630, 966, 1254, 1287, 715, 1, 14, 63, 182, 406, 756, 1218, 1716, 2079, 2002, 1001

Offset: 2

R. J. Mathar, Jul 30 2016

Let F(r,n,d) = binomial(r+d-2,d-1)* ((r-1)*(n-2)+d) /d be the d-dimensional pyramidal numbers. Then A(n,k) = F(k,n,5).

Sum of the n-th antidiagonal is binomial(n+4,7) + binomial(n+4,5) = A055797(n-1). - Mathew Englander, Oct 27 2020

			The array starts in rows n>=2 and columns k>=1 as
   1    5   15   35   70  126  210  330  495   715  1001  1365  1820
   1    6   21   56  126  252  462  792 1287  2002  3003  4368  6188
   1    7   27   77  182  378  714 1254 2079  3289  5005  7371 10556
   1    8   33   98  238  504  966 1716 2871  4576  7007 10374 14924
   1    9   39  119  294  630 1218 2178 3663  5863  9009 13377 19292
   1   10   45  140  350  756 1470 2640 4455  7150 11011 16380 23660
   1   11   51  161  406  882 1722 3102 5247  8437 13013 19383 28028
   1   12   57  182  462 1008 1974 3564 6039  9724 15015 22386 32396
   1   13   63  203  518 1134 2226 4026 6831 11011 17017 25389 36764

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2 <= n <= 150, flattened)
Index to sequences related to polygonal numbers

Cf. Row sums of A080852 (4D), A080851 (3D), A057145 (2D), A077028 (1D).

Cf. A055797.

Table[Binomial[k + 3, 4] + (# - 2)*Binomial[k + 3, 5] &[m - k + 1], {m, 2, 12}, {k, m - 1}] // Flatten (* Michael De Vlieger, Nov 05 2020 *)

A(n+2,k) = Sum_{j=0..k-1} A080852(n,j).

A(n,k) = binomial(k+3,4) + (n-2)*binomial(k+3,5). - Mathew Englander, Oct 27 2020

cp's OEIS Frontend

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