Serhat Bulut - cp's OEIS frontend

A253947 a(n) = 6*binomial(n+1,7).

6, 48, 216, 720, 1980, 4752, 10296, 20592, 38610, 68640, 116688, 190944, 302328, 465120, 697680, 1023264, 1470942, 2076624, 2884200, 3946800, 5328180, 7104240, 9364680, 12214800, 15777450, 20195136, 25632288, 32277696, 40347120, 50086080, 61772832, 75721536

Offset: 6

Serhat Bulut, Jan 20 2015

For a set of integers {1,2,...,n}, a(n) is the sum of the 3 smallest elements of each subset with 6 elements, which is 6*binomial(n+1,7) (for n>=6), hence a(n) = 6*binomial(n+1,7) = 6*A000580(n+1).

			For A={1,2,3,4,5,6,7}, the subsets with 6 elements are {1,2,3,4,5,6}, {1,2,3,4,5,7}, {1,2,3,4,6,7}, {1,2,3,5,6,7}, {1,2,4,5,6,7}, {1,3,4,5,6,7}, and {2,3,4,5,6,7}.
Sum of 3 smallest elements of each subset: a(7) = (1+2+3) + (1+2+3) + (1+2+3) + (1+2+3) + (1+2+4) + (1+3+4) + (2+3+4) = 48 = 6*binomial(7+1,7) = 6*A000580(7+1).

Colin Barker, Table of n, a(n) for n = 6..1000
Serhat Bulut, Oktay Erkan Temizkan, Subset Sum Problem
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000580 (binomial(n, 7)).

[6*Binomial(n+1, 7): n in [6..40]]; // Vincenzo Librandi, Feb 13 2015

Drop[Plus @@ Flatten[Part[#, 1 ;; 3] & /@ Subsets[Range@ #, {6}]] & /@
  Range@ 30, 5] (* Michael De Vlieger, Jan 20 2015 *)
6 Binomial[Range[7, 31], 7] (* Michael De Vlieger, Feb 13 2015, after Alonso del Arte at A253946 *)

Vec(6*x^6/(1-x)^8 + O(x^100)) \\ Colin Barker, Apr 03 2015

a(n) = 6*binomial(n+1,7) = 6*A000580(n+1).
G.f.: 6*x^6 / (1-x)^8. - Colin Barker, Apr 03 2015
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Wesley Ivan Hurt, Sep 03 2022

More terms from Vincenzo Librandi, Feb 13 2015

A253946 a(n) = 6*binomial(n+1, 6).

Original entry on oeis.org

6, 42, 168, 504, 1260, 2772, 5544, 10296, 18018, 30030, 48048, 74256, 111384, 162792, 232560, 325584, 447678, 605682, 807576, 1062600, 1381380, 1776060, 2260440, 2850120, 3562650, 4417686, 5437152, 6645408, 8069424, 9738960, 11686752, 13948704, 16564086

Offset: 5

Serhat Bulut, Jan 20 2015

For a set of integers {1, 2, ..., n}, a(n) is the sum of the 3 smallest elements of each subset with 5 elements, which is 6*C(n+1, 6) (for n >= 5), hence a(n) = 6*C(n+1, 6) = 6 * A000579(n+1).

			For A = {1, 2, 3, 4, 5, 6} the subsets with 5 elements are {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 3, 5, 6}, {1, 2, 4, 5, 6}, {1, 3, 4, 5, 6}, {2, 3, 4, 5, 6}.
The sum of 3 smallest elements of each subset: a(6) = (1 + 2 + 3) + (1 + 2 + 3) + (1 + 2 + 3) + (1 + 2 + 4) + (1 + 3 + 4) + (2 + 3 + 4) = 42 = 6*C(6 + 1, 6) = 6*A000579(6+1).

Colin Barker, Table of n, a(n) for n = 5..1000
Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem, 2015.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000579.

Sixth column of A003506.

[6*Binomial(n+1, 6): n in [5..40]]; // Vincenzo Librandi, Feb 13 2015

A253946:=n->6*binomial(n+1,6): seq(A253946(n), n=5..50); # Wesley Ivan Hurt, Feb 13 2015

Drop[Plus @@ Flatten[Part[#, 1 ;; 3] & /@ Subsets[Range@ #, {5}]] & /@
  Range@ 30, 4] (* Michael De Vlieger, Jan 20 2015 *)
6Binomial[Range[6, 29], 6] (* Alonso del Arte, Feb 05 2015 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{6,42,168,504,1260,2772,5544},40] (* Harvey P. Dale, May 14 2019 *)

Vec(6*x^5/(1-x)^7 + O(x^100)) \\ Colin Barker, Apr 03 2015

a(n) = 6*C(n+1,6) = 6*A000579(n+1).
G.f.: 6*x^5 / (1-x)^7. - Colin Barker, Apr 03 2015
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=5} 1/a(n) = 1/5.
Sum_{n>=5} (-1)^(n+1)/a(n) = 32*log(2) - 661/30. (End)

More terms from Vincenzo Librandi, Feb 13 2015

A253945 a(n) = 6*binomial(n+1,5).

Original entry on oeis.org

6, 36, 126, 336, 756, 1512, 2772, 4752, 7722, 12012, 18018, 26208, 37128, 51408, 69768, 93024, 122094, 158004, 201894, 255024, 318780, 394680, 484380, 589680, 712530, 855036, 1019466, 1208256, 1424016, 1669536, 1947792, 2261952, 2615382, 3011652, 3454542

Offset: 4

Serhat Bulut, Jan 20 2015

For a set of integers {1,2,...,n}, a(n) is the sum of the 3 smallest elements of each subset with 4 elements, which is 6*binomial(n+1,5) for n>=4, hence a(n) = 6*binomial(n+1,5) = 6*A000389(n+1). - Serhat Bulut, Oktay Erkan Temizkan, Jan 20 2015

			For A={1,2,3,4,5}, subsets with 4 elements are {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5,}, {2,3,4,5}.
Sum of 3 smallest elements of each subset: a(5) = (1+2+3) + (1+2+3) + (1+2+4) + (1+3+4) + (2+3+4) = 36 = 6*binomial(5+1,5) = 6*A000389(5+1).

Colin Barker, Table of n, a(n) for n = 4..1000
Serhat Bulut, Oktay Erkan Temizkan, Subset Sum Problem
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A000217,

[6*Binomial(n+1, 5): n in [4..40]]; // Vincenzo Librandi, Feb 13 2015

Drop[Plus @@ Flatten[Part[#, 1 ;; 3] & /@ Subsets[Range@ #, {4}]] & /@
  Range@ 28, 3] (* Michael De Vlieger, Jan 20 2015 *)
6 Binomial[Range[5, 29], 5] (* Michael De Vlieger, Feb 13 2015, after Alonso del Arte at A253946 *)

Vec(6*x^4/(1-x)^6 + O(x^100)) \\ Colin Barker, Apr 03 2015

a(n) = 6*A000389(n+1).
G.f.: 6*x^4 / (1-x)^6. - Colin Barker, Apr 03 2015
a(n) = Sum_{i=1..n-2} A000217(i-1)*A000217(i+1) with a(3)=0. [Bruno Berselli, Jul 20 2015]
E.g.f.: x^4*(5 + x)*exp(x)/20. - G. C. Greubel, Nov 24 2017

More terms from Vincenzo Librandi, Feb 13 2015

A253944 a(n) = 3*binomial(n+1,7).

Original entry on oeis.org

3, 24, 108, 360, 990, 2376, 5148, 10296, 19305, 34320, 58344, 95472, 151164, 232560, 348840, 511632, 735471, 1038312, 1442100, 1973400, 2664090, 3552120, 4682340, 6107400, 7888725, 10097568, 12816144, 16138848, 20173560, 25043040, 30886416, 37860768

Offset: 6

Serhat Bulut, Jan 20 2015

For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 6 elements, which is 3*C(n+1,7) (for n>=6), hence a(n) = 3*C(n+1,7) = 3*A000580(n+1).

			For A={1,2,3,4,5,6,7}, subsets with 6 elements are {1,2,3,4,5,6}, {1,2,3,4,5,7}, {1,2,3,4,6,7}, {1,2,3,5,6,7}, {1,2,4,5,6,7}, {1,3,4,5,6,7}, {2,3,4,5,6,7}.
Sum of 2 smallest elements of each subset:
a(7) = (1+2)+(1+2)+(1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 24 = 3*C(7+1,7) = 3*A000580(7+1).

G. C. Greubel, Table of n, a(n) for n = 6..1000
Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000580.

[3*Binomial(n+1, 7): n in [6..40]]; // Vincenzo Librandi, Feb 13 2015

Drop[Plus @@ Flatten[Part[#, 1 ;; 2] & /@ Subsets[Range@ #, {6}]] & /@
  Range@ 28, 5] (* Michael De Vlieger, Jan 20 2015 *)
3 Binomial[Range[7, 29], 7] (* Michael De Vlieger, Feb 13 2015, after Alonso del Arte at A253946 *)

a(n)=3*binomial(n+1,7) \\ Charles R Greathouse IV, Feb 04 2015

def A253944(n): return 3*binomial(n+1,7)
print([A253944(n) for n in range(6,51)]) # G. C. Greubel, Apr 03 2025

a(n) = 3*C(n+1,7) = 3*A000580(n+1).
a(n) = 3*C(n+1,7) = n*(n^6 - 14*n^5 + 70*n^4 - 140*n^3 + 49*n^2 + 154*n - 120)/1680.
From G. C. Greubel, Apr 03 2025: (Start)
G.f.: 3*x^6/(1-x)^8.
E.g.f.: (3/7!)*x^6*(x+7)*exp(x). (End)

More terms from Vincenzo Librandi, Feb 13 2015

A253943 a(n) = 3*binomial(n+1,6).

Original entry on oeis.org

3, 21, 84, 252, 630, 1386, 2772, 5148, 9009, 15015, 24024, 37128, 55692, 81396, 116280, 162792, 223839, 302841, 403788, 531300, 690690, 888030, 1130220, 1425060, 1781325, 2208843, 2718576, 3322704, 4034712, 4869480, 5843376, 6974352, 8282043, 9787869, 11515140

Offset: 5

Serhat Bulut, Jan 20 2015

nonn

For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 5 elements, which is 3*C(n+1,6) (for n>=5), hence a(n) = 3*C(n+1,6) = 3*A000579(n+1). - Serhat Bulut, Oktay Erkan Temizkan, Jan 20 2015

			For A={1,2,3,4,5,6} subsets with 5 elements are {1,2,3,4,5}, {1,2,3,4,6}, {1,2,3,5,6}, {1,2,4,5,6}, {1,3,4,5,6}, {2,3,4,5,6}.
Sum of 2 smallest elements of each subset:
a(6) = (1+2) + (1+2) + (1+2) + (1+2) + (1+3) + (2+3) = 21 = 3*C(6+1,6) = 3*A000579(6+1).

G. C. Greubel, Table of n, a(n) for n = 5..1000
Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem, 2015.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000579.

[3*Binomial(n+1,6): n in [5..40]]; // Vincenzo Librandi, Feb 13 2015

Drop[Plus @@ Flatten[Part[#, 1 ;; 2] & /@ Subsets[Range@ #, {5}]] & /@
  Range@ 28, 4] (* Michael De Vlieger, Jan 20 2015 *)
3 Binomial[Range[6, 29], 6] (* Michael De Vlieger, Feb 13 2015, after Alonso del Arte at A253946 *)

def A253943(n): return 3*binomial(n+1,6)
print([A253943(n) for n in range(5,51)]) # G. C. Greubel, Apr 03 2025

a(n) = 3*C(n+1,6) = 3*A000579(n+1).
From Amiram Eldar, Jan 09 2022: (Start)
Sum_{n>=5} 1/a(n) = 2/5.
Sum_{n>=5} (-1)^(n+1)/a(n) = 64*log(2) - 661/15. (End)
From G. C. Greubel, Apr 03 2025: (Start)
G.f.: 3*x^5/(1-x)^7.
E.g.f.: (3/6!)*x^5*(x+6)*exp(x). (End)

More terms from Vincenzo Librandi, Feb 13 2015

A253942 a(n) = 3*binomial(n+1, 5).

Original entry on oeis.org

3, 18, 63, 168, 378, 756, 1386, 2376, 3861, 6006, 9009, 13104, 18564, 25704, 34884, 46512, 61047, 79002, 100947, 127512, 159390, 197340, 242190, 294840, 356265, 427518, 509733, 604128, 712008, 834768, 973896, 1130976, 1307691, 1505826, 1727271, 1974024, 2248194

Offset: 4

Serhat Bulut, Jan 20 2015

For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 4 elements, which is 3*binomial(n+1, 5) (for n >= 4), hence a(n) = 3*binomial(n+1, 5) = 3*A000389(n+1).

			For A={1,2,3,4}, the only subset with 4 elements is {1,2,3,4}; sum of 2 minimum elements of this subset: a(4) = 1+2 = 3 = 3*binomial(4+1, 5).
For A={1,2,3,4,5}, the subsets with 4 elements are {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}; sum of 2 smallest elements of each subset: a(5) = (1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 18 = 3*binomial(5+1, 5).

Colin Barker, Table of n, a(n) for n = 4..1000
Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem [Dead link]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389.

[3*Binomial(n+1, 5): n in [4..40]]; // Vincenzo Librandi, Feb 14 2015

a253942[n_] := Drop[Plus @@ Flatten[Part[#, 1 ;; 2] & /@ Subsets[Range@ #, {4}]] & /@ Range@ n, 3]; a253942[28] (* Michael De Vlieger, Jan 20 2015 *)
Table[3 Binomial[n + 1, 5], {n, 4, 35}] (* Vincenzo Librandi, Feb 14 2015 *)

a(n) = 3*binomial(n+1, 5); \\ Michel Marcus, Jan 20 2015

Vec(3*x^4/(x-1)^6 + O(x^100)) \\ Colin Barker, Jan 20 2015

a(n) = 3*A000389(n+1).
a(n) = (n-3)*(n-2)*(n-1)*n*(1+n)/40. - Colin Barker, Jan 20 2015
G.f.: 3*x^4 / (x-1)^6. - Colin Barker, Jan 20 2015
E.g.f.: x^4*(x+5)*exp(x)/40. - G. C. Greubel, Nov 25 2017
a(n) = Sum_{k=3..n-1} A050534(k). - Ivan N. Ianakiev, Oct 08 2023

cp's OEIS Frontend

User: Serhat Bulut

A253947 a(n) = 6*binomial(n+1,7).

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A253946 a(n) = 6*binomial(n+1, 6).

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A253945 a(n) = 6*binomial(n+1,5).

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A253944 a(n) = 3*binomial(n+1,7).

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A253943 a(n) = 3*binomial(n+1,6).

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A253942 a(n) = 3*binomial(n+1, 5).

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