A185508 -id:A185508 - cp's OEIS frontend

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

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

Clark Kimberling, Feb 03 2011

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

[n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

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 *)

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

Edited by Clark Kimberling, Feb 25 2023

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.

A264750 Number of sequences of 5 throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the 5th throw.

Original entry on oeis.org

5, 29, 99, 259, 574, 1134, 2058, 3498, 5643, 8723, 13013, 18837, 26572, 36652, 49572, 65892, 86241, 111321, 141911, 178871, 223146, 275770, 337870, 410670, 495495, 593775, 707049, 836969, 985304, 1153944, 1344904, 1560328, 1802493, 2073813, 2376843, 2714283

Offset: 5

Louis Rogliano, Nov 23 2015

Sequence gives the second column of A185508. [Bruno Berselli, Nov 24 2015]

Number of 5-tuples (t_1, ..., t_5) with 1 <= t_j <= n, Sum_{j <= 4} t_j < n and Sum_{j<=5} t_j >= n. - Robert Israel, Nov 25 2015

			From _Jon E. Schoenfield_, Nov 26 2015: (Start)
For n=5, the a(5) = 5 sequences (i.e., quintuples or 5-tuples) are {1,1,1,1,1}, {1,1,1,1,2}, {1,1,1,1,3}, {1,1,1,1,4} and {1,1,1,1,5}. (Each of the first four throws must be a 1; otherwise, the sum of the throws would reach or exceed 5 before the 5th throw.)
For n=6, each of the quintuples must have four throws whose sum is less than 6, followed by a fifth throw that brings the sum to at least 6, so the a(6) = 29 quintuples are the 5 quintuples {1,1,1,1,t_5} where t_5 is any value in 2..6 and the four sets of 6 quintuples {1,1,1,2,t_5}, {1,1,2,1,t_5}, {1,2,1,1,t_5} and {2,1,1,1,t_5} where t_5 is any value in 1..6. (End)

Colin Barker, Table of n, a(n) for n = 5..1000
Louis Rogliano, Sequence A264750
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000096 (k=2), A051925 (k=3), A215862 (k=4).

Cf. A185508.

[(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120: n in [5..40]]; // Vincenzo Librandi, Nov 24 2015

A264750:=n->(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120: seq(A264750(n), n=5..50); # Wesley Ivan Hurt, Nov 24 2015

f[n_, k_] := Module[
{i, total = 0, part, perm},
part = IntegerPartitions[n, {k}];
perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}],      1];
For[i = 1, i <= Length[perm], i++,    total += n + 1 - perm[[i, k]]    ];
Return[total];   ]
And the sequences are obtained by:
h[k_] := Table[f[i, k], {i, k, number_of_terms_wanted}]
Table[(n - 4) (n - 3) (n - 2) (n - 1) (4 n + 5)/120, {n, 5, 40}] (* Bruno Berselli, Nov 24 2015 *)

Vec(x^5*(5-x)/(1-x)^6 + O(x^100)) \\ Colin Barker, Nov 23 2015

for(n=5, 40, print1((n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120", ")); \\ Bruno Berselli, Nov 24 2015

[(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120 for n in (5..40)] # Bruno Berselli, Nov 24 2015

From Colin Barker, Nov 23 2015: (Start)
a(n) = (n - 4)*(n - 3)*(n - 2)*(n - 1)*(4*n + 5)/120.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>10.
G.f.: x^5*(5 - x) / (1 - x)^6. (End)

Offset changed by Robert Israel, Nov 25 2015

Formulae, b-file adapted to the new offset and definition rephrased by the Editors of the OEIS, Nov 26 2015

cp's OEIS Frontend

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

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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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A264750 Number of sequences of 5 throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the 5th throw.

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