A254870 Seventh partial sums of fourth powers (A000583).

1, 23, 221, 1355, 6239, 23465, 75803, 217373, 566150, 1361802, 3063502, 6508450, 13159666, 25481470, 47493274, 85567222, 149553199, 254336185, 421956275, 684451365, 1087616985, 1695917535, 2598828765, 3918943275, 5822229660, 8530902276, 12339433068

Offset: 1

Author

Luciano Ancora, Feb 17 2015

Keywords

• nonn
• easy

Examples

			Second differences:   2, 14,  50,  110,  194,   302, ...   A120328(2k+1)
First differences:    1, 15,  65,  175,  369,   671, ...   A005917
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The fourth powers:    1, 16,  81,  256,  625,  1296, ...   A000583
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First partial sums:   1, 17,  98,  354,  979,  2275, ...   A000538
Second partial sums:  1, 18, 116,  470, 1449,  3724, ...   A101089
Third partial sums:   1, 19, 135,  605, 2054,  5778, ...   A101090
Fourth partial sums:  1, 20, 155,  760, 2814,  8592, ...   A101091
Fifth partial sums:   1, 21, 176,  936, 3750, 12342, ...   A254681
Sixth partial sums:   1, 22, 198, 1134, 4884, 17226, ...   A254470
Seventh partial sums: 1, 23, 221, 1355, 6239, 23465, ...   (this sequence)
		

Links

• 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).

Crossrefs

Cf. A000538, A000583, A005917, A101089, A101090, A101091, A254681, A254470, A254869, A254871, A254872.

Programs

Magma

[n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(7+2*n)*(7 +42*n+6*n^2)/19958400: n in [1..30]]; // Vincenzo Librandi, Feb 19 2015

Mathematica

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 2 n)((7 + 42 n + 6 n^2)/19958400), {n, 24}] (* or *)
CoefficientList[Series[(1 + 11 x + 11 x^2 + x^3)/(- 1 + x)^12, {x, 0, 23}], x]

PARI

vector(50,n,n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(7 + 42*n + 6*n^2)/19958400) \\ Derek Orr, Feb 19 2015

Formula

G.f.: (x + 11*x^2 + 11*x^3 + x^4)/(- 1 + x)^12.

a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(7 + 42*n + 6*n^2)/19958400.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^4.