A022523 -id:A022523 - cp's OEIS frontend

A254646 Fourth partial sums of seventh powers (A001015).

1, 132, 2709, 26432, 168126, 804552, 3136014, 10459968, 30856839, 82407052, 202678203, 465069696, 1005729452, 2066218896, 4058958828, 7664805504, 13974953853, 24692818836, 42415687153, 71020845504, 116186669130, 186085891160, 292296070170, 450981236160, 684408934755

Offset: 1

Luciano Ancora, Feb 05 2015

			First differences:   1, 127, 2059, 14197,  61741, ...  (A022523)
----------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384,  78125, ...  (A001015)
----------------------------------------------------------------------
First partial sums:  1, 129, 2316, 18700,  96825, ...  (A000541)
Second partial sums: 1, 130, 2446, 21146, 117971, ...  (A250212)
Third partial sums:  1, 131, 2577, 23723, 141694, ...  (A254641)
Fourth partial sums: 1, 132, 2709, 26432, 168126, ...  (this sequence)

Luciano Ancora, Table of n, a(n) for n = 1..1000
Luciano Ancora, Partial sums of m-th powers with Faulhaber polynomials
Luciano Ancora, Pascal’s triangle and recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000541, A001015, A022523, A250212, A254641, A254645.

List([1..30], n-> Binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198); # G. C. Greubel, Aug 28 2019

[Binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198: n in [1..30]]; // G. C. Greubel, Aug 28 2019

seq(binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198, n=1..30); # G. C. Greubel, Aug 28 2019

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (48 - 100 n - 89 n^2 + 160 n^3 + 140 n^4 + 36 n^5 + 3 n^6)/23760, {n, 20}] (* or *)
Accumulate[Accumulate[Accumulate[Accumulate[Range[20]^7]]]] (* or *)
CoefficientList[Series[(1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/(- 1 + x)^12, {x, 0, 19}], x]

a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(48-100*n-89*n^2+160*n^3+140*n^4 +36*n^5+3*n^6)/23760 \\ Charles R Greathouse IV, Oct 07 2015

[binomial(n+4,5)*(3*(n+2)^6 -40*(n+2)^4 +151*(n+2)^2 -108)/198 for n in (1..30)] # G. C. Greubel, Aug 28 2019

G.f.: x*(1 +120*x +1191*x^2 +2416*x^3 +1191*x^4 +120*x^5 +x^6)/(1-x)^12.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(48 - 100*n - 89*n^2 + 160*n^3 + 140*n^4 + 36*n^5 + 3*n^6)/23760.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + n^7.

A255181 Third differences of seventh powers (A001015).

Original entry on oeis.org

1, 125, 1806, 10206, 35406, 92526, 201726, 388206, 682206, 1119006, 1738926, 2587326, 3714606, 5176206, 7032606, 9349326, 12196926, 15651006, 19792206, 24706206, 30483726, 37220526, 45017406, 53980206, 64219806, 75852126, 88998126, 103783806, 120340206

Offset: 0

Luciano Ancora, Mar 18 2015

			Third differences:   1, 125, 1806, 10206, 35406, ...  (this sequence)
Second differences:  1, 126, 1932, 12138, 47544, ...  (A255177)
First differences:   1, 127, 2059, 14197, 61741, ...  (A022523)
---------------------------------------------------------------------
The seventh powers:  1, 128, 2187, 16384, 78125, ...  (A001015)
---------------------------------------------------------------------

Lucuano Ancora, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000541, A001015, A022523, A255177.

[1,125] cat [42*(3-10*n+15*n^2-10*n^3+5*n^4): n in [2..30]]; // Vincenzo Librandi, Mar 18 2015

Join[{1, 125}, Table[42 (3 - 10 n + 15 n^2 - 10 n^3 + 5 n^4), {n, 2, 30}]]

G.f.: (1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1 - x)^5.
a(n) = 42*(3 - 10*n + 15*n^2 - 10*n^3 + 5*n^4) for n>1, a(0)=1, a(1)=125.
a(n) = A255177(n)-A255177(n-1). - R. J. Mathar, Jul 16 2015

Edited by Bruno Berselli, Mar 19 2015

A343237 Triangle T obtained from the array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0, by reading antidiagonals upwards.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 15, 19, 7, 1, 1, 31, 65, 37, 9, 1, 1, 63, 211, 175, 61, 11, 1, 1, 127, 665, 781, 369, 91, 13, 1, 1, 255, 2059, 3367, 2101, 671, 127, 15, 1, 1, 511, 6305, 14197, 11529, 4651, 1105, 169, 17, 1

Offset: 0

Wolfdieter Lang, May 10 2021

This is the row reversed version of the triangle A047969(n, m). The corresponding array A047969 is a(n, k) = A(k, n) (transposed of array A).

A(n-1, k-1) = k^n - (k-1)^n gives the number of n-digit numbers with digits from K = {1, 2, 3, ..., k} such that any digit from K, say k, appears at least once. Motivated by a comment in A005061 by Enrique Navarrete, the instance k=4 for n >= 1 (the column 3 in array A), and the d = 3 (sub)-diagonal sequence of T for m >= 0.

			The array A begins:
n\k  0  1   2    3     4     5     6      7      8      9 ...
-------------------------------------------------------------
0:   1  1   1    1     1     1     1      1      1      1 ...
1:   1  3   5    7     9    11    13     15     17     19 ...
2:   1  7  19   37    61    91   127    169    217    271 ...
3:   1 15  65  175   369   671  1105   1695   2465   3439 ...
4:   1 31 211  781  2101  4651  9031  15961  26281  40951 ...
5:   1 63 665 3367 11529 31031 70993 144495 269297 468559 ...
...
The triangle T begins:
n\m   0    1     2     3     4     5    6    7   8  9 10 ...
-------------------------------------------------------------
0:    1
1:    1    1
2:    1    3     1
3:    1    7     5     1
4:    1   15    19     7     1
5:    1   31    65    37     9     1
6:    1   63   211   175    61    11    1
7:    1  127   665   781   369    91   13    1
8:    1  255  2059  3367  2101   671  127   15   1
9:    1  511  6305 14197 11529  4651 1105  169  17  1
10:   1 1023 19171 58975 61741 31031 9031 1695 217 19  1
...
Combinatorial interpretation (cf. A005061 by _Enrique Navarrete_)
The three digits numbers with digits from K ={1, 2, 3, 4} having at least one 4 are:
j=1 (one 4): 114, 141, 411; 224, 242, 422; 334, 343, 433; 124, 214, 142, 241, 412, 421; 134, 314, 143, 341, 413, 431; 234, 243, 423. That is,  3*3 + 3!*3 = 27 = binomial(3, 1)*(4-1)^(3-1) = 3*3^2;
j=2 (twice 4):  144, 414, 441;  244, 424, 442;  344, 434, 443; 3*3 = 9 = binomial(3, 2)*(4-1)^(3-2) = 3*3;
j=3 (thrice 4) 444; 1 = binomial(3, 3)*(4-1)^(3-3).
Together: 27 + 9 + 1 = 37 = A(2, 3) = T(5, 3).

Cf. A005061, A008292, A047969 (reversed), A045531 (central diagonal), A047970 (row sums of triangle).
Row sequences of array A (nexus numbers): A000012, A005408, A003215, A005917(k+1), A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528.
Column sequences of array A: A000012, A000225(n+1), A001047(n+1), A005061(n+1), A005060(n+1), A005062(n+1), A016169(n+1), A016177(n+1), A016185(n+1), A016189(n+1), A016195(n+1), A016197(n+1).

egf := exp(exp(x)*y + x)*(exp(x)*y - y + 1): ser := series(egf, x, 12):
cx := n -> series(n!*coeff(ser, x, n), y, 12):
Arow := n -> seq(k!*coeff(cx(n), y, k), k=0..9):
for n from 0 to 5 do Arow(n) od; # Peter Luschny, May 10 2021

A[n_, k_] := (k + 1)^(n + 1) - k^(n + 1); Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 10 2021 *)

Array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0.
A(n-1, k-1) = Sum_{j=1} binomial(n, j)*(k-1)^(n-j) = Sum_{j=0} binomial(n, j)*(k-1)^(n-j) - (k-1)^n = (1+(k-1))^n - (k-1)^n = k^n - (k-1)^n (from the combinatorial comment on A(n-1, k-1) above).
O.g.f. row n of array A: RA(n, x) = P(n, x)/(1 - x)^n, with P(n, x) = Sum_{m=0..n} A008292(n+1, m+1)*x^m, (the Eulerian number triangle  A008292 has offset 1) for n >= 0. (See the Oct 26 2008 comment in A047969 by Peter Bala). RA(n, x) = polylog(-(n+1), x)*(1-x)/x. (For P(n, x) see the formula by Vladeta Jovovic, Sep 02 2002.)
E.g.f. of e.g.f.s of the rows of array A: EE(x, y) = exp(x)*(1 + y*(exp(x) - 1))*exp(y*exp(x)), that is A(n, k) = [y^k/k!][x^n/n!] EE(x, y).
Triangle T(n, m) = A(n-m, m) = (m+1)^(n-m+1) - m^(n-m+1), n >= 0, m = 0, 1, ..., n.
E.g.f.: -(exp(x)-1)/(x*exp(x)*y-x). - Vladimir Kruchinin, Nov 02 2022

A341050 Cube array read by upward antidiagonals ignoring zero and empty terms: T(n, k, r) is the number of n-ary strings of length k, containing r consecutive 0's.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 5, 8, 1, 1, 3, 1, 5, 8, 1, 7, 21, 19, 1, 1, 3, 1, 5, 8, 1, 7, 21, 20, 1, 9, 40, 81, 43, 1, 1, 3, 1, 5, 8, 1, 7, 21, 20, 1, 9, 40, 81, 47, 1, 11, 65, 208, 295, 94, 1, 1, 3, 1, 5, 8, 1, 7, 21, 20, 1, 9, 40, 81, 48, 1, 11, 65, 208, 297, 107, 1, 13, 96, 425, 1024, 1037, 201

Offset: 2

Robert P. P. McKone, Feb 04 2021

			For n = 5, k = 6 and r = 4, there are 65 strings: {000000, 000001, 000002, 000003, 000004, 000010, 000011, 000012, 000013, 000014, 000020, 000021, 000022, 000023, 000024, 000030, 000031, 000032, 000033, 000034, 000040, 000041, 000042, 000043, 000044, 010000, 020000, 030000, 040000, 100000, 100001, 100002, 100003, 100004, 110000, 120000, 130000, 140000, 200000, 200001, 200002, 200003, 200004, 210000, 220000, 230000, 240000, 300000, 300001, 300002, 300003, 300004, 310000, 320000, 330000, 340000, 400000, 400001, 400002, 400003, 400004, 410000, 420000, 430000, 440000}
The first seven slices of the tetrahedron (or pyramid) are:
-----------------Slice 1-----------------
  1
-----------------Slice 2-----------------
    1
  1  3
-----------------Slice 3-----------------
      1
    1  3
  1  5  8
-----------------Slice 4-----------------
        1
      1  3
    1  5   8
  1  7  21  19
-----------------Slice 5-----------------
          1
        1  3
      1  5   8
    1  7  21  20
  1  9  40  81  43
-----------------Slice 6-----------------
              1
           1    3
        1    5     8
      1   7    21    20
    1   9   40    81    47
  1  11  65   208   295   94
-----------------Slice 7-----------------
                 1
              1     3
           1     5     8
         1    7     21    20
      1    9    40     81      48
    1   11   65    208     297     107
  1  13   96   425    1024    1037    201

Robert P. P. McKone, Antidiagonals n = 2..50, flattened

Cf. A340156 (r=2), A340242 (r=3).
Cf. A008466 (n=2, r=2), A186244 (n=3, r=2), A050231 (n=2, r=3), A231430 (n=3, r=3).
Cf. A005408, A003215, A005917, A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528, A022529, A022530, A022531, A022532, A022533, A022534, A022535, A022536, A022537, A022538, A022539, A022540 (k=x, r=1, where x is the x-th Nexus Number).
Cf. A000567 [(k=4, r=2),(k=5, r=3),(k=6, r=4),...,(k=x, r=x-2)].
Cf. A103532 [(k=6, r=3),(k=7, r=4),(k=8, r=5),...,(k=x, r=x-3)].

m[r_, n_] := Normal[With[{p = 1/n}, SparseArray[{Band[{1, 2}] -> p, {i_, 1} /; i <= r -> 1 - p, {r + 1, r + 1} -> 1}]]]; T[n_, k_, r_] := MatrixPower[m[r, n], k][[1, r + 1]]*n^k; DeleteCases[Transpose[PadLeft[Reverse[Table[T[n, k, r], {k, 2, 8}, {r, 2, k}, {n, 2, r}], 2]], 2 <-> 3], 0, 3] // Flatten

A259907 Fifth differences of 7th powers (A001015).

Original entry on oeis.org

1, 123, 1557, 6719, 16800, 31920, 52080, 77280, 107520, 142800, 183120, 228480, 278880, 334320, 394800, 460320, 530880, 606480, 687120, 772800, 863520, 959280, 1060080, 1165920, 1276800, 1392720, 1513680, 1639680, 1770720, 1906800, 2047920, 2194080, 2345280, 2501520, 2662800

Offset: 0

Kolosov Petro, Jul 07 2015

			1 128 2187 16384 78125 279936 823543 2097152 4782969 (seventh powers)
1 127 2059 14197 61741 201811 543607 1273609 2685817 (first differences)
1 126 1932 12138 47544 140070 341796  730002 1412208 (second differences)
1 125 1806 10206 35406  92526 201726  388206  682206 (third differences)
1 124 1681  8400 25200  57120 109200  186480  294000 (fourth differences)
1 123 1557  6719 16800  31920  52080   77280  107520 (here)

John H. Conway and Richard K. Guy, The Book of Numbers. New York: Springer-Verlag, pp. 30-32, 1996.
Kiran Parulekar. Amazing Properties of Squares and Their Calculations. Kiran Anil Parulekar, 2012.
Bag, Amulya Kumar (1966). "Binomial theorem in ancient India". Indian J. History Sci 1 (1): 68-74.
Ronald Graham and Donald Knuth, Patashnik, Oren (1994). "(5) Binomial Coefficients". Concrete Mathematics (2nd ed.). Addison Wesley. pp. 153-256.

R. J. Mathar, Table of n, a(n) for n = 0..79
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001015, A022523, A255177, A255181.

[1,123,1557,6719] cat [840*(3*n^2-9*n+8): n in [4..40]]; // Bruno Berselli, Jul 16 2015

Join[{1, 123, 1557, 6719}, Table[840 (3 n^2 - 9 n + 8), {n, 4, 40}]]

[1,123,1557,6719]+[840*(3*n^2-9*n+8) for n in (4..40)] # Bruno Berselli, Jul 16 2015

G.f.: (1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1 - x)^3.
a(n) = 840*(3*n^2 - 9*n + 8) for n>3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6. - Vincenzo Librandi, Jul 08 2015

Edited by Editors of the OEIS, Jul 16 2015

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A254646 Fourth partial sums of seventh powers (A001015).

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A255181 Third differences of seventh powers (A001015).

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A343237 Triangle T obtained from the array A(n, k) = (k+1)^(n+1) - k^(n+1), n, k >= 0, by reading antidiagonals upwards.

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A341050 Cube array read by upward antidiagonals ignoring zero and empty terms: T(n, k, r) is the number of n-ary strings of length k, containing r consecutive 0's.

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A259907 Fifth differences of 7th powers (A001015).

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