A144112 -id:A144112 - cp's OEIS frontend

A115262 Correlation triangle for n+1.

1, 2, 2, 3, 5, 3, 4, 8, 8, 4, 5, 11, 14, 11, 5, 6, 14, 20, 20, 14, 6, 7, 17, 26, 30, 26, 17, 7, 8, 20, 32, 40, 40, 32, 20, 8, 9, 23, 38, 50, 55, 50, 38, 23, 9, 10, 26, 44, 60, 70, 70, 60, 44, 26, 10, 11, 29, 50, 70, 85, 91, 85, 70, 50, 29, 11

Offset: 0

Paul Barry, Jan 18 2006

This sequence (formatted as a square array) gives the counts of all possible squares in an m X n rectangle. For example, 11 = 8 (1 X 1 squares) + 3 (2 X 2 square) in 4 X 2 rectangle. - Philippe Deléham, Nov 26 2009
From Clark Kimberling, Feb 07 2011: (Start)
Also the accumulation array of min{n,k}, when formatted as a rectangle.
This is the accumulation array of the array M=A003783 given by M(n,k)=min{n,k}; see A144112 for the definition of accumulation array.
The accumulation array of A115262 is A185957. (End)
From Clark Kimberling, Dec 22 2011: (Start)
As a square matrix, A115262 is the self-fusion matrix of A000027 (1,2,3,4,...).  See A193722 for the definition of fusion and A202673 for characteristic polynomials associated with A115622. (End)

			Triangle begins
  1;
  2,  2;
  3,  5,  3;
  4,  8,  8,  4;
  5, 11, 14, 11,  5;
  6, 14, 20, 20, 14,  6;
  ...
When formatted as a square matrix:
  1,  2,  3,  4,  5, ...
  2,  5,  8, 11, 14, ...
  3,  8, 14, 20, 26, ...
  4, 11, 20, 30, 40, ...
  5, 14, 26, 40, 55, ...
  ...

Cf. A000027, A202673, A271916.
For the triangular version: row sums are A001752. Diagonal sums are A097701. T(2n,n) is A000330(n+1).
Diagonals (1,5,...): A000330 (square pyramidal numbers),
diagonals (2,8,...): A007290,
diagonals (3,11,...): A051925,
diagonals (4,14,...): A159920,
antidiagonal sums: A001752.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
(* Clark Kimberling, Dec 22 2011 *)

Let f(m,n) = m*(m-1)*(3*n-m-1)/6. This array is (with a different offset) the infinite square array read by antidiagonals U(m,n) = f(n,m) if m < n, U(m,n) = f(m,n) if m <= n. See A271916. - N. J. A. Sloane, Apr 26 2016
G.f.: 1/((1-x)^2*(1-x*y)^2*(1-x^2*y)).
Number triangle T(n, k) = Sum_{j=0..n} [j<=k]*(k-j+1)[j<=n-k]*(n-k-j+1).
T(2n,n) - T(2n,n+1) = n+1.

A185508 Third accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 5, 6, 16, 29, 21, 41, 89, 99, 56, 91, 219, 295, 259, 126, 182, 469, 705, 755, 574, 252, 336, 910, 1470, 1765, 1645, 1134, 462, 582, 1638, 2786, 3605, 3780, 3206, 2058, 792, 957, 2778, 4914, 6706, 7595, 7266, 5754, 3498, 1287, 1507, 4488, 8190, 11634, 13916, 14406, 12894, 9690, 5643, 2002, 2288, 6963, 13035, 19110, 23814, 26068, 25284, 21510, 15510, 8723, 3003, 3367, 10439, 19965, 30030, 38640, 44100

Offset: 1

Clark Kimberling, Jan 29 2011

See A144112 (and A185506) for the definition of accumulation array (aa).

Sequence is aa(aa(aa(A000027))).

			Northwest corner:
   1    5   16   41   91  182
   6   29   89  219  469  910
  21   99  295  705 1470 2786
  56  259  755 1765 3605 6706

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000027, A185506, A185507, A185509.

Cf. A000389 (column 1), A257199 (row 1).

h[n_,k_]:=k(k+1)(k+2)n(n+1)(n+2)*(4n^2+(5k+23)n+4k^2+3k+41)/2880;
TableForm[Table[h[n,k],{n,1,10},{k,1,15}]]
Table[h[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

{h(n,k) = k*(k+1)*(k+2)*n*(n+1)*(n+2)*(4*n^2+(5*k+23)*n +4*k^2 +3*k + 41)/2880}; for(n=1,10, for(k=1,n, print1(h(k, n-k+1), ", "))) \\ G. C. Greubel, Nov 23 2017

T(n,k) = F*(4n^2 + (5k+23)n + 4k^2 + 3k+41), where F = k(k+1)(k+2)n(n+1)(n+2)/2880.

A185874 Second accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 6, 11, 10, 10, 21, 26, 20, 15, 34, 48, 50, 35, 21, 50, 76, 90, 85, 56, 28, 69, 110, 140, 150, 133, 84, 36, 91, 150, 200, 230, 231, 196, 120, 45, 116, 196, 270, 325, 350, 336, 276, 165, 55, 144, 248, 350, 435, 490, 504, 468, 375, 220, 66, 175, 306, 440, 560, 651, 700, 696, 630, 495, 286, 78, 209, 370, 540, 700, 833, 924, 960, 930, 825, 638, 364, 91, 246, 440, 650, 855, 1036, 1176, 1260, 1275, 1210, 1056, 806, 455, 105, 286, 516, 770, 1025, 1260, 1456, 1596, 1665, 1650, 1540, 1326, 1001, 560

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain: A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
.   1,   3,   6,   10,   15,   21,   28,   36,   45,   55, ...
.   4,  11,  21,   34,   50,   69,   91,  116,  144,  175, ...
.  10,  26,  48,   76,  110,  150,  196,  248,  306,  370, ...
.  20,  50,  90,  140,  200,  270,  350,  440,  540,  650, ...
.  35,  85, 150,  230,  325,  435,  560,  700,  855, 1025, ...
.  56, 133, 231,  350,  490,  651,  833, 1036, 1260, 1505, ...
.  84, 196, 336,  504,  700,  924, 1176, 1456, 1764, 2100, ...
. 120, 276, 468,  696,  960, 1260, 1596, 1968, 2376, 2820, ...
. 165, 375, 630,  930, 1275, 1665, 2100, 2580, 3105, 3675, ...
. 220, 495, 825, 1210, 1650, 2145, 2695, 3300, 3960, 4675, ...
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A144112, A051340, A141419, A185875, A185876.
Row 1 to 5: A000217, A115056, 2*A140096, 10*A000096, 5*A059845.
Column 1 to 3: A000292, A051925, A267370 and 3*A005581.
Main diagonal: A117066.

f[n_, k_] := (1/12)*k*n*(1 + n)*(1 + 3*k + 2*n);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten

T(n,k) = k*n*(n+1)*(2*n+3*k+1)/12 for k>=1, n>=1.

Edited by Bruno Berselli, Jan 14 2016

A144216 C(m,2)+C(n,2), m>=1, n>=1: a rectangular array R read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 6, 4, 4, 6, 10, 7, 6, 7, 10, 15, 11, 9, 9, 11, 15, 21, 16, 13, 12, 13, 16, 21, 28, 22, 18, 16, 16, 18, 22, 28, 36, 29, 24, 21, 20, 21, 24, 29, 36, 45, 37, 31, 27, 25, 25, 27, 31, 37, 45, 55, 46, 39, 34, 31, 30, 31, 34, 39, 46, 55, 66, 56, 48, 42, 38, 36, 36, 38

Offset: 1

Clark Kimberling, Sep 14 2008

This is the accumulation array (as defined at A144112) of the weight array A144217.

As a triangular array read by rows (0; 1, 1; 3, 2, 3; 6, 4, 4, 6; ...), T(n,j) = (1/2)n(n+1-2j)+j(j-1) (1<=j<=n) is the sum of the distances from the vertex j of the path graph 1-2-...-n to all the other vertices. Example: T(4,2)=4 because in the path 1-2-3-4 the distances from vertex 2 to the vertices 1, 2, 3, 4 are 1, 0, 1, 2, respectively; 1+0+1+2=4.

			   0,  1,  3,  6, 10, 15, 21, 28, 36, 45, ...
   1,  2,  4,  7, 11, 16, 22, 29, 37, 46, ...
   3,  4,  6,  9, 13, 18, 24, 31, 39, 48, ...
   6,  7,  9, 12, 16, 21, 27, 34, 42, 51, ...
  10, 11, 13, 16, 20, 25, 31, 38, 46, 55, ...
  15, 16, 18, 21, 25, 30, 36, 43, 51, 60, ...
  21, 22, 24, 27, 31, 36, 42, 49, 57, 66, ...
  28, 29, 31, 34, 38, 43, 49, 56, 64, 73, ...
  36, 37, 39, 42, 46, 51, 57, 64, 72, 81, ...
  45, 46, 48, 51, 55, 60, 66, 73, 81, 90, ...
R(2,4) = binomial(2,2) + binomial(4,2) = 1 + 6 = 7.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened).

Cf. A144217.

T := proc (n, j) if j <= n then (1/2)*n*(n+1-2*j)+j*(j-1) else 0 end if end proc: for n to 12 do seq(T(n, j), j = 1 .. n) end do; # yields sequence in triangular form

Table[n(n-m-1)+m(m+1)/2,{m,15},{n,m}] (* Paolo Xausa, Dec 21 2023 *)

R(m,n) = (m(m-1)+n(n-1))/2.

The sum of the terms in the upper left r X r submatrix is Sum_{n=1..r} Sum_{m=1..r} R(n,m) = A112742(r). - J. M. Bergot, Jun 18 2013

A144225 Bordered Pascal's triangle in rectangular format.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 6, 4, 1, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 0, 1, 8, 28, 56, 70, 56, 28, 8, 1, 0, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45

Offset: 1

Clark Kimberling, Sep 15 2008

This is the weight array (defined at A144112) of Pascal's rectangle - that is, Pascal's triangle A007318 formatted as a rectangle.

			Northwest corner:
1 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7
0 1 3 6 10 15 21 28
0 1 4 10 20 35 56

A007318, A144112.

After deleting row 1, (1 0 0 0 ...) and column 1, (1 0 0 0 ...), the remaining array is given by R(m,n)=C(m+n-2,m-1). This "Pascal rectangle" is the accumulation array of A144225.

A185506 Accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 7, 11, 10, 14, 23, 26, 20, 25, 42, 51, 50, 35, 41, 70, 88, 94, 85, 56, 63, 109, 140, 156, 155, 133, 84, 92, 161, 210, 240, 250, 237, 196, 120, 129, 228, 301, 350, 375, 374, 343, 276, 165, 175, 312, 416, 490, 535, 550, 532, 476, 375, 220, 231, 415, 558, 664, 735, 771, 770, 728, 639, 495, 286

Offset: 1

Clark Kimberling, Jan 29 2011

Suppose that R={R(n,k) : n>=1, k>=1} is a rectangular array. The accumulation array of R is given by T(n,k) = Sum_{R(i,j): 1<=i<=n, 1<=j<=k}. (See A144112.)

The formula for the integer T(n,k) has denominator 12. The 2nd, 3rd, and 4th accumulation arrays of A000027 have formulas in which the denominators are 144, 2880, and 86400, respectively; see A185507, A185508, and A185509.

			The natural number array A000027 starts with
  1, 2,  4,  7, ...
  3, 5,  8, 12, ...
  6, 9, 13, 18, ...
  ...
T(n,k) is the sum of numbers in the rectangle with corners at (1,1) and (n,k) of A000027, so that a corner of T is as follows:
   1,  3,   7,  14,  25,  41
   4, 11,  23,  42,  70, 109
  10, 26,  51,  88, 140, 210
  20, 50,  94, 156, 240, 350
  35, 85, 155, 250, 375, 535

G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals, flattened

Cf. A000027, A185507, A185508, A185509.

Cf. A004006 (row 1), A000292 (col 1), A051925 (col 2), A185505 (1st diagonal).

f[n_,k_]:=k*n*(2n^2+3(k+1)*n+2k^2-3k+5)/12;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k*n*(2*n^2 + 3*(k+1)*n + 2*k^2 - 3*k + 5)/12.

A185507 Second accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 4, 5, 11, 19, 15, 25, 49, 55, 35, 50, 105, 136, 125, 70, 91, 200, 280, 300, 245, 126, 154, 350, 515, 600, 575, 434, 210, 246, 574, 875, 1075, 1125, 1001, 714, 330, 375, 894, 1400, 1785, 1975, 1925, 1624, 1110, 495, 550, 1335, 2136, 2800, 3220, 3325, 3080, 2496, 1650, 715, 781, 1925, 3135, 4200, 4970, 5341, 5250, 4680, 3675, 2365, 1001, 1079, 2695, 4455, 6075, 7350, 8134, 8330, 7890, 6825, 5225, 3289, 1365, 1456, 3679, 6160, 8525, 10500, 11886, 12544, 12390, 11400, 9625, 7216, 4459, 1820, 1925, 4914, 8320, 11660

Offset: 1

Clark Kimberling, Jan 29 2011

See A144112 (and A185506) for the definition of accumulation array (aa).

Sequence is aa(aa(A000027)).

			Northwest corner:
   1,   4,  11,   25,   50,   91,  154
   5,  19,  49,  105,  200,  350,  574
  15,  55, 136,  280,  515,  875, 1400
  35, 125, 300,  600, 1075, 1785, 2800
  70, 245, 575, 1125, 1975, 3220, 4970

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A006522 (row 1), A000332 (column 1).

Cf. A000027, A185506, A185508, A185509.

g[n_,k_]:=k*n(k+1)(n+1)(3n^2+(4k+11)n+3k^2-k+16)/144;
TableForm[Table[g[n,k],{n,1,10},{k,1,15}]]
Table[g[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k*n*(k+1)*(n+1)*(3*n^2 + (4*k+11)*n + 3*k^2 - k + 16)/144.

A185509 Fourth accumulation array, T, of the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 6, 7, 22, 41, 28, 63, 146, 161, 84, 154, 406, 561, 476, 210, 336, 966, 1526, 1631, 1176, 462, 672, 2058, 3556, 4361, 3976, 2562, 924, 1254, 4032, 7434, 9996, 10486, 8568, 5082, 1716, 2211, 7392, 14322, 20580, 23716, 22344, 16842, 9372, 3003, 3718, 12837, 25872, 39102, 48216, 49980, 43512, 30822, 16302, 5005, 6006, 21307, 44352, 69762, 90552, 100548, 96432, 79002, 53262, 27027, 8008, 9373, 34034, 72787, 118272, 159852

Offset: 1

Clark Kimberling, Jan 29 2011

See A144112 (and A185506) for the definition of rectangular sum array (aa).

Sequence is aa(aa(aa(aa(A000027)))).

			Northwest corner:
1.....6....22....63...154
7....41...146...406...966
28..161...561..1526..3556
84..476..1631..4361..9996

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000027, A185506, A185507, A185508.

Cf. A000579 (column 1), A257200 (row 1).

u[n_,k_]:=k(k+1)(k+2)(k+3)n(n+1)(n+2)(n+3)(5n^2+(6k+39)n+5k^2+9k+86)/86400
TableForm[Table[u[n,k],{n,1,10},{k,1,15}]]
Table[u[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = F*(5*n^2 + (6*k + 39)*n + 5*k^2 + 9*k + 86), where

F = k*(k+1)*(k+2)*(k+3)*n*(n+1)*(n+2)*(n+3)/86400.

A185875 Third accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 4, 5, 10, 19, 15, 20, 46, 55, 35, 35, 90, 130, 125, 70, 56, 155, 250, 290, 245, 126, 84, 245, 425, 550, 560, 434, 210, 120, 364, 665, 925, 1050, 980, 714, 330, 165, 516, 980, 1435, 1750, 1820, 1596, 1110, 495, 220, 705, 1380, 2100, 2695, 3010, 2940, 2460, 1650, 715, 286, 935, 1875, 2940, 3920, 4606, 4830, 4500, 3630, 2365, 1001, 364, 1210, 2475, 3975, 5460, 6664, 7350, 7350, 6600, 5170

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
   1,   4,  10,  20,  35
   5,  19,  46,  90, 155
  15,  55, 130, 250, 425
  35, 125, 290, 550, 925

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A051340, A141419, A144112, A185875, A185876.

Row 1: A000292; Column 1: A000332.

f[n_,k_]:= k*(1+k)*n*(1+n)*(2+n)*(5+4*k+3*n)/144;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = (3*n+4*k+5)*C(k,2)*C(n,3)/12, k>=1, n>=1.

A185876 Fourth accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 5, 6, 15, 29, 21, 35, 85, 99, 56, 70, 195, 285, 259, 126, 126, 385, 645, 735, 574, 252, 210, 686, 1260, 1645, 1610, 1134, 462, 330, 1134, 2226, 3185, 3570, 3150, 2058, 792, 495, 1770, 3654, 5586, 6860, 6930, 5670, 3498, 1287, 715, 2640, 5670, 9114, 11956, 13230, 12390, 9570, 5643, 2002, 1001, 3795, 8415, 14070

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
   1,   5,  15,   35,   70
   6,  29,  85,  195,  385
  21,  99, 285,  645, 1260
  56, 259, 735, 1645, 3185

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A051340, A141419, A144112, A185874, A185875.

Row 1: A000332, column 1: A000389.

f[n_,k_]:=k(1+k)n(1+n)(2+n)(5+4k+3n)/144;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185875 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185876 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185876 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = (4*n+5*k+11)*C(k+2,3)*C(n+4,4)/20, k>=1, n>=1.

cp's OEIS Frontend

A115262 Correlation triangle for n+1.

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A185508 Third accumulation array, T, of the natural number array A000027, by antidiagonals.

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A185874 Second accumulation array of A051340, by antidiagonals.

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A144216 C(m,2)+C(n,2), m>=1, n>=1: a rectangular array R read by antidiagonals.

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A144225 Bordered Pascal's triangle in rectangular format.

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A185506 Accumulation array, T, of the natural number array A000027, by antidiagonals.

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A185507 Second accumulation array, T, of the natural number array A000027, by antidiagonals.

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A185509 Fourth accumulation array, T, of the natural number array A000027, by antidiagonals.

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