Luciano Ancora - cp's OEIS frontend

A259914 Staircase path through the array P(n,k) of the k-th partial sums of cubes (A000578).

1, 9, 10, 46, 57, 203, 272, 846, 1200, 3432, 5082, 13728, 21021, 54483, 85696, 215254, 346086, 848198, 1388900, 3337236, 5549786, 13119614, 22108704, 51557260, 87885070, 202588830, 348817770, 796117860, 1382941125, 3129153795

Offset: 1

Luciano Ancora, Jul 08 2015

The term "stepped path" in the name field is the same used in A001405 and A259775.

			The array begins:
[1], [9],  36,   100,   225,    441,  ...  A000537
1,  [10], [46],  146,   371,    812,  ...  A024166
1,   11,  [57], [203],  574,   1386,  ...  A101094
1,   12,   69,  [272], [846],  2232,  ...  A101097
1,   13,   82,   354, [1200], [3432], ...  A101102
1,   14,   96,   450,  1650,  [5082], ...  A254469

Cf. A000537, A024166, A101094, A101097, A101102, A254469.

Table[DifferenceRoot[Function[{a, n},
         {(-650880 - 1496112*n - 1426512*n^2 - 722164*n^3 - 204716*n^4 - 30812*n^5 - 1924*n^6)*a[n] + (-56736 - 140412*n - 132006*n^2 - 58114*n^3 - 12090*n^4 - 962*n^5)*a[1 + n] + (78624 + 229884*n + 273800*n^2 + 167579*n^3 + 54567*n^4 + 8665*n^5 + 481*n^6)*a[2 + n] == 0, a[1] == 1, a[2] == 9}]][n], {n, 30}]

Conjecture: 2*(n+7)*(145672*n^2-236343*n+123525)*a(n) +(-78613*n^3-794662*n^2+327391*n+20220)*a(n-1) +2*(-582688*n^3-1889455*n^2-2148719*n-832650)*a(n-2) +4*(n-1)*(78613*n^2+133361*n+64050)*a(n-3) = 0. - R. J. Mathar, Jul 16 2015

A259775 Stepped path in P(k,n) array of k-th partial sums of squares (A000290).

Original entry on oeis.org

1, 5, 6, 20, 27, 77, 112, 294, 450, 1122, 1782, 4290, 7007, 16445, 27456, 63206, 107406, 243542, 419900, 940576, 1641486, 3640210, 6418656, 14115100, 25110020, 54826020, 98285670, 213286590, 384942375

Offset: 1

Luciano Ancora, Jul 05 2015

The term "stepped path" in the name field is the same used in A001405.

Interleaving of terms of the sequences A220101 and A129869. - Michel Marcus, Jul 05 2015

			The array of k-th partial sums of squares begins:
[1], [5],  14,   30,    55,     91,  ...  A000330
1,   [6], [20],  50,   105,    196,  ...  A002415
1,    7,  [27], [77],  182,    378,  ...  A005585
1,    8,   35, [112], [294],   672,  ...  A040977
1,    9,   44,  156,  [450], [1122], ...  A050486
1,   10,   54,  210,   660,  [1782], ...  A053347
This is essentially A110813 without its first two columns.

Cf. A000330, A002415, A005585, A040977, A050486, A053347.

Table[DifferenceRoot[Function[{a, n}, {(-9168 - 14432*n - 8412*n^2 - 2152*n^3 - 204*n^4)*a[n] +(-1332 - 1902*n - 792*n^2 - 102*n^3)*a[1 + n] + (2100 + 3884*n + 2493*n^2 + 640*n^3 + 51*n^4)*a[2 + n] == 0, a[1] == 1 , a[2] == 5}]][n], {n, 29}]

Conjecture: -(n+5)*(13*n-11)*a(n) +(8*n^2+39*n-35)*a(n-1) +2*(26*n^2+48*n+25)*a(n-2) -4*(8*n+5)*(n-1)*a(n-3)=0. - R. J. Mathar, Jul 16 2015

A259348 a(n) = n^3 - 8.

Original entry on oeis.org

-8, -7, 0, 19, 56, 117, 208, 335, 504, 721, 992, 1323, 1720, 2189, 2736, 3367, 4088, 4905, 5824, 6851, 7992, 9253, 10640, 12159, 13816, 15617, 17568, 19675, 21944, 24381, 26992, 29783, 32760, 35929, 39296, 42867, 46648, 50645, 54864, 59311, 63992

Offset: 0

Luciano Ancora, Jun 24 2015

The cubic number sequence whose geometrical arrangement loses all vertices: this is a figurate number represented by a cubic lattice of n^3 equispaced points without vertices.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[n^3 - 8: n in [0..40]]; // Vincenzo Librandi, Jun 25 2015

Table[n^3 - 8, {n, 0, 40}] (* Vincenzo Librandi, Jun 25 2015 *)
LinearRecurrence[{4,-6,4,-1},{-8,-7,0,19},50] (* Harvey P. Dale, Sep 25 2017 *)

G.f.: (-8 + 25*x - 20*x^2 + 9*x^3)/(1-x)^4. - Vincenzo Librandi, Jun 25 2015

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 25 2015

First term -8 added from Vincenzo Librandi, Jun 25 2015

A257838 Main diagonal of iterated partial sums array of Fibonacci numbers (starting with the first partial sums).

Original entry on oeis.org

0, 1, 4, 16, 63, 247, 967, 3785, 14820, 58060, 227612, 892926, 3505386, 13770404, 54129602, 212904952, 837885495, 3299264407, 12997784803, 51230474669, 202014314769, 796928589755, 3145066003589, 12416625685891, 49037912997003, 193734379979677, 765632076098287, 3026670770970925, 11968378998073935

Offset: 0

Luciano Ancora, May 10 2015

The array used here starts in row n=0 with the first partial sums of A000045. The array which starts with the Fibonacci numbers in row k=0 is shown in A136431. The diagonal of that array is given in A176085. - Wolfdieter Lang, Jun 03 2015

			This sequence is the main diagonal of the following array (see the comment and Example field of A136431):
0, 1, 2,  4,  7,  12, ...  A000071
0, 1, 3,  7, 14,  26, ...  A001924
0, 1, 4, 11, 25,  51, ...  A014162
0, 1, 5, 16, 41,  92, ...  A014166
0, 1, 6, 22, 63, 155, ...  A053739
0, 1, 7, 29, 92, 247, ...  A053295

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000045, A000071, A001924, A014162, A014166, A136431, A176085.

Table[DifferenceRoot[Function[{a, n},{(2*n + 4*n^2)*a[n] + (2 + 7*n + 15*n^2)*a[1 + n] + (8 - 6*n - 8*n^2)*a[2 + n] + (-2 + n + n^2)*a[3 + n] == 0, a[1] == 0, a[2] == 1, a[3] == 4, a[4] == 16}]][n], {n, 30}]

a(n):=sum(binomial(2*n-k,n-k)*fib(k),k,0,n); /* Vladimir Kruchinin, Oct 09 2016 */

x='x+O('x^50); concat([0], Vec(-(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x))) \\ G. C. Greubel, Apr 08 2017

a(n) = F^{n+1}(n), n >= 0, with the k-th iterated partial sum F^{k} of the Fibonacci number A000045. - Wolfdieter Lang, Jun 03 2015
Conjecture: n*(n-3)*a(n) +2*(-4*n^2+13*n-6)*a(n-1) +(15*n^2-53*n+48)*a(n-2) +2*(2*n-3)*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 10 2015
G.f.: -(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x). - Vladimir Kruchinin, Oct 09 2016
a(n) = Sum_{k=0..n} binomial(2*n-k,n-k)*F(k), where F(k) = A000045(k). - Vladimir Kruchinin, Oct 09 2016
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 09 2016

Name edited by Wolfdieter Lang, Jun 03 2015

A257449 a(n) = 75(2^n - 1) - 4n^3 - 18n^2 - 52n.

Original entry on oeis.org

1, 17, 99, 373, 1115, 2901, 6907, 15509, 33483, 70405, 145451, 296997, 601819, 1213493, 2439195, 4893301, 9804587, 19630629, 39286603, 78602885, 157240251, 314520277, 629086139, 1258224213, 2516507275, 5033080901, 10066236267, 20132555749, 40265204123

Offset: 1

Luciano Ancora, Apr 23 2015

See the first comment of A257448.

			This sequence provides the antidiagonal sums of the array:
1, 16,  81, 256,  625,  1296, ...   A000583
1, 17,  98, 354,  979,  2275, ...   A000538
1, 18, 116, 470, 1449,  3724, ...   A101089
1, 19, 135, 605, 2054,  5778, ...   A101090
1, 20, 155, 760, 2814,  8592, ...   A101091
1, 21, 176, 936, 3750, 12342, ...   A254681
...
See also A254681 (Example field).

Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Cf. A000225, A000670, A050488, A208744, A257448, A257450.

[75*(2^n-1)-4*n^3-18*n^2-52*n: n in [1..30]]; // Vincenzo Librandi, Apr 24 2015

Table[75 (2^n - 1) - 4 n^3 - 18 n^2 - 52 n, {n, 30}]

G.f.: -x*(1 + x)*(1 + 10*x + x^2)/((-1 + x)^4*(-1 + 2*x)).

a(n) = 6*a(n-1) -14*a(n-2) +16*a(n-3) -9*a(n-4) +2*a(n-5) for n>5.

A257450 a(n) = 541(2^n - 1) - 5n^4 - 30n^3 - 130n^2 - 375*n.

Original entry on oeis.org

1, 33, 277, 1335, 4771, 14193, 37417, 90795, 207871, 456693, 974437, 2036655, 4195771, 8558073, 17337697, 34964595, 70300471, 141070653, 282727837, 566179575, 1133243251, 2267556033, 4536394777, 9074315835, 18150434671, 36302985093, 72608437717, 145219736895

Offset: 1

Luciano Ancora, Apr 23 2015

See the first comment of A257448.

			This sequence provides the antidiagonal sums of the array:
1, 32, 243, 1024,  3125,  7776, ...   A000584
1, 33, 276, 1300,  4425, 12201, ...   A000539
1, 34, 310, 1610,  6035, 18236, ...   A101092
1, 35, 345, 1955,  7990, 26226, ...   A101099
1, 36, 381, 2336, 10326, 36552, ...   A254644
1, 37, 418, 2754, 13080, 49632, ...   A254682
...
See also A254682 (Example field).

Index entries for linear recurrences with constant coefficients, signature (7,-20,30,-25,11,-2).

Cf. A000225, A000670, A050488, A208744, A257448, A257449.

[541*(2^n-1)-5*n^4-30*n^3-130*n^2-375*n: n in [1..30]]; // Vincenzo Librandi, Apr 24 2015

Table[541 (2^n - 1) - 5 n^4 - 30 n^3 - 130 n^2 - 375 n, {n, 30}]
LinearRecurrence[{7,-20,30,-25,11,-2},{1,33,277,1335,4771,14193},30] (* Harvey P. Dale, Dec 24 2018 *)

G.f.: x*(1+26*x+66*x^2+26*x^3+x^4)/(-1+x)^5*(-1+2*x).

a(n) = 7*a(n-1) -20*a(n-2) +30*a(n-3) -25*a(n-4) +11*a(n-5) -2*a(n-6) for n>6.

A257448 a(n) = 13(2^n - 1) - 3n^2 - 9*n.

Original entry on oeis.org

1, 9, 37, 111, 283, 657, 1441, 3051, 6319, 12909, 26149, 52695, 105859, 212265, 425161, 851043, 1702903, 3406725, 6814477, 13630095, 27261451, 54524289, 109050097, 218101851, 436205503, 872412957, 1744828021, 3489658311, 6979319059, 13958640729

Offset: 1

Luciano Ancora, Apr 23 2015

These numbers belong to a family of sequences obtained as follows:
. A000225:   1*(2^n-1);
. A050488:   3*(2^n-1) - 2*n;
.    a(n):  13*(2^n-1) - 3*n^2 -  9*n;
. A257449:  75*(2^n-1) - 4*n^3 - 18*n^2 -  52*n;
. A257450: 541*(2^n-1) - 5*n^4 - 30*n^3 - 130*n^2 - 375*n,
where the sequence 1, 3, 13, 75, 541, ... is A000670 (after the first term), and A208744 gives the triangle of coefficients:
2;
3, 9;
4, 18, 52;
5, 30, 130, 375;
6, 45, 260, 1125, 3246;
7, 63, 455, 2625, 11361, 32781, etc.
Also, the antidiagonal sums in the array are given by the formula (6*n^2 + 6*k*n + (k-1)*k)*(k+n)!/((k+3)!*(n-1)!) for k = 0, 1, 2, 3, 4, ... (see Example field).

			By the second comment, the array begins (antidiagonals in A046902):
k=0: 1,  8, 27,  64,  125,  216, ...  A000578
k=1: 1,  9, 36, 100,  225,  441, ...  A000537
k=2: 1, 10, 46, 146,  371,  812, ...  A024166
k=3: 1, 11, 57, 203,  574, 1386, ...  A101094
k=4: 1, 12, 69, 272,  846, 2232, ...  A101097
k=5: 1, 13, 82, 354, 1200, 3432, ...  A101102
k=6: 1, 14, 96, 450, 1650, 5082, ...  A254469
...
See also A254469 (Example field).

Index entries for linear recurrences with constant coefficients, signature (5, -9, 7, -2).

Cf. A000225, A000670, A050488, A208744, A257449, A257450.

[13*(2^n-1)-3*n^2-9*n: n in [1..30]]; // Vincenzo Librandi, Apr 24 2015

Table[13 (2^n - 1) - 3 n^2 - 9n, {n, 30}]
CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^3*(1 - 2 x)), {x, 0, 30}], x] (* Michael De Vlieger, Nov 14 2016 *)

G.f.: x*(1+4*x+x^2)/((1-x)^3*(1-2*x)).

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>4. - Ray Chandler, Jul 25 2015

Edited by Bruno Berselli, Apr 28 2015

A257200 a(n) = n(n+1)(n+2)(n+3)(n^2+3*n+26)/720.

Original entry on oeis.org

1, 6, 22, 63, 154, 336, 672, 1254, 2211, 3718, 6006, 9373, 14196, 20944, 30192, 42636, 59109, 80598, 108262, 143451, 187726, 242880, 310960, 394290, 495495, 617526, 763686, 937657, 1143528, 1385824, 1669536, 2000152, 2383689, 2826726, 3336438, 3920631, 4587778, 5347056, 6208384, 7182462

Offset: 1

Luciano Ancora, Apr 18 2015

Antidiagonal sums of the array of 4-dimensional solid numbers shown in Table 3 of Sardelis and Valahas paper (see also Example field).
See A257199 (second comment) for the general formula of this type of numbers: the sequence correspond to the case j = 4.
Binomial transform of (1, 5, 11, 14, 11, 5, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

			Array in Comments begins:
1,  5, 15,  35,  70, 126, 210,  330, ...
1,  6, 20,  50, 105, 196, 336,  540, ...
1,  7, 25,  65, 140, 266, 462,  750, ...
1,  8, 30,  80, 175, 336, 588,  960, ...
1,  9, 35,  95, 210, 406, 714, 1170, ...
1, 10, 40, 110, 245, 476, 840, 1380, ...

D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070 [math.GM], 2008.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A257199, A257201.

See A080852 for another version of the array.

[n*(n+1)*(n+2)*(n+3)*(n^2+3*n+26)/720: n in [1..40]]; // Vincenzo Librandi, Apr 18 2015

Table[n (n + 1) (n + 2) (n + 3) (n^2 + 3n + 26)/720, {n, 40}]

first(m)=vector(m,i,i*(i+1)*(i+2)*(i+3)*(i^2+3*i+26)/720) \\ Anders Hellström, Aug 26 2015

Vec(x*(-1 + x - x^2)/(-1 + x)^7 + O(x^40)) \\ Michel Marcus, Aug 27 2015

G.f.: x*(-1 + x - x^2)/(-1 + x)^7.

A257199 a(n) = n(n+1)(n+2)(n^2+2n+17)/120.

Original entry on oeis.org

1, 5, 16, 41, 91, 182, 336, 582, 957, 1507, 2288, 3367, 4823, 6748, 9248, 12444, 16473, 21489, 27664, 35189, 44275, 55154, 68080, 83330, 101205, 122031, 146160, 173971, 205871, 242296, 283712, 330616, 383537, 443037, 509712, 584193, 667147, 759278, 861328, 974078

Offset: 1

Luciano Ancora, Apr 18 2015

Antidiagonal sums of the array of pyramidal numbers shown in Table 2 of Sardelis and Valahas paper (see A261720).
This is the case j = 3 of (n^2 + (j-1)*n + (j+1)^2 + 1)*binomial(n+j-1, j)/((j+1)*(j+2)), where j is the space dimension: a(n) = (n^2+2*n+17)*binomial(n+2,3)/20.
The sequence is the binomial transform of (1, 4, 7, 7, 4, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070 [math.GM], 2008.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A257200, A257201.

For another version of the array, see A080851.

[n*(n+1)*(n+2)*(n^2+2*n+17)/120: n in [1..40]]; // Vincenzo Librandi, Apr 18 2015

Table[n (n + 1) (n + 2) (n^2 + 2n + 17)/120, {n, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,5,16,41,91,182},40] (* Harvey P. Dale, Mar 18 2018 *)

G.f.: x*(1 - x + x^2)/(1 - x)^6.

A257201 a(n) = n(n+1)(n+2)(n+3)(n+4)(n^2+4n+37)/5040.

Original entry on oeis.org

1, 7, 29, 92, 246, 582, 1254, 2508, 4719, 8437, 14443, 23816, 38012, 58956, 89148, 131784, 190893, 271491, 379753, 523204, 710930, 953810, 1264770, 1659060, 2154555, 2772081, 3535767, 4473424, 5616952, 7002776, 8672312, 10672464, 13056153, 15882879, 19219317, 23139948, 27727726, 33074782, 39283166, 46465628

Offset: 1

Luciano Ancora, Apr 18 2015

Antidiagonal sums of the array of 5-dimensional solid numbers (see Example field).
See A257199 (second comment) for the general formula of this type of numbers: the sequence correspond to the case j = 5.
The sequence is the binomial transform of (1, 6, 16, 25, 25, 16, 6, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

			Array in Comments begins:
  1,  6, 21,  56, 126,  252,  462,  792, 1287, 2002, ...
  1,  7, 27,  77, 182,  378,  714, 1254, 2079, 3289, ...
  1,  8, 33,  98, 238,  504,  966, 1716, 2871, 4576, ...
  1,  9, 39, 119, 294,  630, 1218, 2178, 3663, 5863, ...
  1, 10, 45, 140, 350,  756, 1470, 2640, 4455, 7150, ...
  1, 11, 51, 161, 406,  882, 1722, 3102, 5247, 8437, ...
  1, 12, 57, 182, 462, 1008, 1974, 3564, 6039, 9724, ...
  ...

D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070 [math.GM], 2008.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A257199, A257200.

[n*(n+1)*(n+2)*(n+3)*(n+4)*(n^2+4*n+37)/5040: n in [1..40]]; // Vincenzo Librandi, Apr 18 2015

Table[n (n + 1) (n + 2) (n + 3) (n + 4) (n^2 + 4n + 37)/5040, {n, 40}]

G.f.: x*(1 - x + x^2)/(1 - x)^8.

cp's OEIS Frontend

User: Luciano Ancora

A259914 Staircase path through the array P(n,k) of the k-th partial sums of cubes (A000578).

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A259775 Stepped path in P(k,n) array of k-th partial sums of squares (A000290).

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A259348 a(n) = n^3 - 8.

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A257838 Main diagonal of iterated partial sums array of Fibonacci numbers (starting with the first partial sums).

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A257449 a(n) = 75*(2^n - 1) - 4*n^3 - 18*n^2 - 52*n.

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A257450 a(n) = 541*(2^n - 1) - 5*n^4 - 30*n^3 - 130*n^2 - 375*n.

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A257448 a(n) = 13*(2^n - 1) - 3*n^2 - 9*n.

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A257449 a(n) = 75(2^n - 1) - 4n^3 - 18n^2 - 52n.

A257450 a(n) = 541(2^n - 1) - 5n^4 - 30n^3 - 130n^2 - 375*n.

A257448 a(n) = 13(2^n - 1) - 3n^2 - 9*n.

A257200 a(n) = n(n+1)(n+2)(n+3)(n^2+3*n+26)/720.

A257199 a(n) = n(n+1)(n+2)(n^2+2n+17)/120.

A257201 a(n) = n(n+1)(n+2)(n+3)(n+4)(n^2+4n+37)/5040.