cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

Original entry on oeis.org

1, 7, 25, 62, 125, 221, 357, 540, 777, 1075, 1441, 1882, 2405, 3017, 3725, 4536, 5457, 6495, 7657, 8950, 10381, 11957, 13685, 15572, 17625, 19851, 22257, 24850, 27637, 30625, 33821, 37232, 40865, 44727, 48825, 53166, 57757, 62605, 67717, 73100, 78761, 84707, 90945, 97482, 104325, 111481, 118957, 126760, 134897, 143375
Offset: 1

Views

Author

Clark Kimberling, Feb 03 2011

Keywords

Comments

This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
1....2.....4.....7...11...16...22...29...
3....5.....8....12...17...23...30...38...
6....9....13....18...24...31...39...48...
10...14...19....25...32...40...49...59...
15...20...26....33...41...50...60...71...
21...27...34....42...51...61...72...84...
28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787. Deleting the main diagonal and then summing give A185787. Analogous treatments to the left of the main diagonal give A100182 and A101165. Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
row 1: A000124
row 2: A022856
row 3: A016028
row 4: A145018
row 5: A077169
col 1: A000217
col 2: A000096
col 3: A034856
col 4: A055998
col 5: A046691
col 6: A052905
col 7: A055999
diag. (1,5,...) ...... A001844
diag. (2,8,...) ...... A001105
diag. (4,12,...)...... A046092
diag. (7,17,...)...... A056220
diag. (11,23,...) .... A132209
diag. (16,30,...) .... A054000
diag. (22,38,...) .... A090288
diag. (3,9,...) ...... A058331
diag. (6,14,...) ..... A051890
diag. (10,20,...) .... A005893
diag. (15,27,...) .... A097080
diag. (21,35,...) .... A093328
antidiagonal sums: (1,5,15,34,...)=A006003=partial sums of A002817.
Let S(n,k) denote the n-th partial sum of column k. Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Weight array of T: A144112
Accumulation array of T: A185506
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509

Links

Crossrefs

Cf. A000027, A185788, A100182, A101165, A079824.

Programs

  • Magma
    [n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012
  • Mathematica
    f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n,k],{n,1,k}];
Factor[s[k]]
Table[s[k],{k,1,70}]  (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)

Formula

a(n)=n*(7*n^2-6*n+5)/6.
G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012

Extensions

Edited by Clark Kimberling, Feb 25 2023

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

Views

Author

Clark Kimberling, Jan 29 2011

Keywords

Comments

See A144112 (and A185506) for the definition of accumulation array (aa).
Sequence is aa(aa(aa(A000027))).

Examples

			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

Links

Crossrefs

Cf. A000027, A185506, A185507, A185509.
Cf. A000389 (column 1), A257199 (row 1).

Programs

  • Mathematica
    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
  • PARI
    {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

Formula

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.

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

Views

Author

Clark Kimberling, Jan 29 2011

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A000027, A185507, A185508, A185509.
Cf. A004006 (row 1), A000292 (col 1), A051925 (col 2), A185505 (1st diagonal).

Programs

  • Mathematica
    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

Formula

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

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

Views

Author

Clark Kimberling, Jan 29 2011

Keywords

Comments

See A144112 (and A185506) for the definition of rectangular sum array (aa).
Sequence is aa(aa(aa(aa(A000027)))).

Examples

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

Links

Crossrefs

Cf. A000027, A185506, A185507, A185508.
Cf. A000579 (column 1), A257200 (row 1).

Programs

  • Mathematica
    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

Formula

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.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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