cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A373912 Number of compositions of 7*n into parts 6 and 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1718, 3448, 6556, 12121, 22509, 43453, 89150, 193823, 436304, 989759, 2219064, 4869285, 10434412, 21900170, 45297211, 93054446, 191371581, 396480142, 830227401, 1756883373, 3746468095, 8017653633, 17151612398
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373908, A373909, A373910, A373911.
Cf. A099099, A099131, A107025, A373913.
Cf. A017847.

Programs

  • PARI
    a(n) = sum(k=0, n\6, binomial(n+k, n-6*k));

Formula

a(n) = A017847(7*n).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^6/(1 - x)^6).

A371125 Number of compositions of 6*n into parts 1 and 6.

Original entry on oeis.org

1, 2, 9, 43, 196, 882, 3970, 17887, 80608, 363254, 1636944, 7376591, 33241289, 149795989, 675029164, 3041899638, 13707783053, 61771701389, 278363253873, 1254394801761, 5652708454881, 25472931513057, 114789263420590, 517277526141329, 2331019740675071
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A052544, A055988, A369836, A373907, A373890.
Cf. A107025, A373904, A373905, A373906.
Cf. A005708.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k, n-k));

Formula

a(n) = A005708(6*n).
a(n) = Sum_{k=0..n} binomial(n+5*k,n-k).
a(n) = 7*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: 1/(1 - x - x/(1 - x)^5).

A373908 Number of compositions of 7*n into parts 2 and 7.

Original entry on oeis.org

1, 1, 2, 9, 38, 136, 452, 1495, 5031, 17114, 58282, 198032, 671856, 2278870, 7731892, 26238839, 89047335, 302191369, 1025487338, 3479970844, 11809261583, 40074827170, 135994407483, 461498426696, 1566098800484, 5314568565096, 18035031128780, 61202027710656
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373909, A373910, A373911, A373912.
Cf. A109961, A369840, A373904.
Cf. A369813.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+5*k, n-2*k));

Formula

a(n) = A369813(7*n).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+5*k,n-2*k).
a(n) = 7*a(n-1) - 20*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^2/(1 - x)^6).

A373909 Number of compositions of 7*n into parts 3 and 7.

Original entry on oeis.org

1, 1, 1, 2, 9, 37, 122, 346, 913, 2398, 6515, 18317, 52226, 148408, 417810, 1168085, 3258813, 9103828, 25488736, 71462437, 200406479, 561770980, 1573939555, 4408629727, 12348599802, 34592601763, 96916209910, 271537125048, 760777555986, 2131439888257
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373908, A373910, A373911, A373912.
Cf. A369814.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n+4*k, n-3*k));

Formula

a(n) = A369814(7*n).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+4*k,n-3*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 36*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^3/(1 - x)^6).

A373910 Number of compositions of 7*n into parts 4 and 7.

Original entry on oeis.org

1, 1, 1, 1, 2, 9, 37, 121, 332, 808, 1837, 4113, 9497, 23091, 58462, 150129, 382810, 960520, 2373982, 5816480, 14230964, 34948927, 86295036, 213973997, 531470618, 1319411997, 3270186871, 8091796123, 20002405065, 49435009494, 122222402392, 302354237393
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373908, A373909, A373911, A373912.
Cf. A369815.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n+3*k, n-4*k));

Formula

a(n) = A369815(7*n).
a(n) = Sum_{k=0..floor(n/4)} binomial(n+3*k,n-4*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 34*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^4/(1 - x)^6).

A373911 Number of compositions of 7*n into parts 5 and 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 794, 1732, 3553, 7116, 14501, 31078, 70607, 166922, 399315, 946121, 2197582, 4998597, 11188280, 24835641, 55117511, 123036293, 276976136, 628285812, 1431723937, 3265884047, 7436635822, 16880558594, 38196652951, 86238054374
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A373907, A373908, A373909, A373910, A373912.
Cf. A369816.

Programs

  • Mathematica
    LinearRecurrence[{7,-21,35,-35,22,-7,1},{1,1,1,1,1,2,9},40] (* Harvey P. Dale, Oct 19 2024 *)
  • PARI
    a(n) = sum(k=0, n\5, binomial(n+2*k, n-5*k));

Formula

a(n) = A369816(7*n).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+2*k,n-5*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 22*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^5/(1 - x)^6).

A373928 Number of compositions of 7*n-2 into parts 1 and 7.

Original entry on oeis.org

1, 7, 35, 168, 819, 4025, 19796, 97315, 478304, 2350860, 11554621, 56791883, 279136551, 1371977475, 6743373646, 33144194898, 162906243014, 800696596250, 3935484773527, 19343207491818, 95073338508548, 467292702057555, 2296779231936167, 11288844908179562
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A099253, A373907, A373929, A373930, A373931, A373932.
Cf. A005709, A369805, A369837.

Programs

  • Mathematica
    a[n_]:= n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*HypergeometricPFQ[{1-n, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6, (10+n)/6}, {6/7, 8/7, 9/7, 10/7, 11/7, 12/7}, -6^6/7^7]/120; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
LinearRecurrence[{8,-21,35,-35,21,-7,1},{1,7,35,168,819,4025,19796},40] (* Harvey P. Dale, Jul 28 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+4+6*k, n-1-k));

Formula

a(n) = A005709(7*n-2).
a(n) = Sum_{k=0..n} binomial(n+4+6*k,n-1-k).
a(n) = 8*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).
G.f.: x*(1-x)/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*hypergeom([1-n, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6, (10+n)/6], [6/7, 8/7, 9/7, 10/7, 11/7, 12/7], -6^6/7^7)/120. - Stefano Spezia, Jun 23 2024

A373929 Number of compositions of 7*n-3 into parts 1 and 7.

Original entry on oeis.org

1, 6, 28, 133, 651, 3206, 15771, 77519, 380989, 1872556, 9203761, 45237262, 222344668, 1092840924, 5371396171, 26400821252, 129762048116, 637790353236, 3134788177277, 15407722718291, 75730131016730, 372219363549007, 1829486529878612, 8992065676243395
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A099253, A373907, A373928, A373930, A373931, A373932.
Cf. A005709, A369806.

Programs

  • Mathematica
    a[n_]:=n*(1 + n)*(2 + n)*(3 + n)*HypergeometricPFQ[{1-n, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6}, {5/7, 6/7, 8/7, 9/7, 10/7, 11/7}, -6^6/7^7]/24; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+3+6*k, n-1-k));

Formula

a(n) = A005709(7*n-3).
a(n) = Sum_{k=0..n} binomial(n+3+6*k,n-1-k).
a(n) = 8*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).
G.f.: x*(1-x)^2/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*(3 + n)*hypergeom([1-n, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6], [5/7, 6/7, 8/7, 9/7, 10/7, 11/7], -6^6/7^7)/24. - Stefano Spezia, Jun 23 2024

A373930 Number of compositions of 7*n-4 into parts 1 and 7.

Original entry on oeis.org

1, 5, 22, 105, 518, 2555, 12565, 61748, 303470, 1491567, 7331205, 36033501, 177107406, 870496256, 4278555247, 21029425081, 103361226864, 508028305120, 2496997824041, 12272934541014, 60322408298439, 296489232532277, 1457267166329605, 7162579146364783
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A099253, A373907, A373928, A373929, A373931, A373932.
Cf. A005709, A369807.

Programs

  • Mathematica
    a[n_]:=n*(1 + n)*(2 + n)*HypergeometricPFQ[{1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6}, {4/7, 5/7, 6/7, 8/7, 9/7, 10/7}, -6^6/7^7]/6; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+2+6*k, n-1-k));

Formula

a(n) = A005709(7*n-4).
a(n) = Sum_{k=0..n} binomial(n+2+6*k,n-1-k).
a(n) = 8*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).
G.f.: x*(1-x)^3/((1-x)^7 - x).
a(n) = n*(1 + n)*(2 + n)*hypergeom([1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6], [4/7, 5/7, 6/7, 8/7, 9/7, 10/7], -6^6/7^7)/6. - Stefano Spezia, Jun 23 2024

A373931 Number of compositions of 7*n-5 into parts 1 and 7.

Original entry on oeis.org

1, 4, 17, 83, 413, 2037, 10010, 49183, 241722, 1188097, 5839638, 28702296, 141073905, 693388850, 3408058991, 16750869834, 82331801783, 404667078256, 1988969518921, 9775936716973, 48049473757425, 236166824233838, 1160777933797328, 5705311980035178
Offset: 1

Views

Author

Seiichi Manyama, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A099253, A373907, A373928, A373929, A373930, A373932.
Cf. A005709, A369808.

Programs

  • Mathematica
    a[n_]:=n*(1 + n)*HypergeometricPFQ[{1-n,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6}, {3/7, 4/7, 5/7, 6/7, 8/7, 9/7}, -6^6/7^7]/2; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+1+6*k, n-1-k));

Formula

a(n) = A005709(7*n-5).
a(n) = Sum_{k=0..n} binomial(n+1+6*k,n-1-k).
a(n) = 8*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).
G.f.: x*(1-x)^4/((1-x)^7 - x).
a(n) = n*(1 + n)*hypergeom([1-n,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6], [3/7, 4/7, 5/7, 6/7, 8/7, 9/7], -6^6/7^7)/2. - Stefano Spezia, Jun 23 2024
Showing 1-10 of 12 results. Next

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