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 1-6 of 6 results.

A096957 Fourth column (m=3) of (1,6)-Pascal triangle A096956.

Original entry on oeis.org

6, 19, 40, 70, 110, 161, 224, 300, 390, 495, 616, 754, 910, 1085, 1280, 1496, 1734, 1995, 2280, 2590, 2926, 3289, 3680, 4100, 4550, 5031, 5544, 6090, 6670, 7285, 7936, 8624, 9350, 10115, 10920, 11766, 12654, 13585, 14560, 15580, 16646, 17759, 18920
Offset: 0

Views

Author

Wolfdieter Lang, Aug 13 2004

Keywords

Comments

If Y is a 6-subset of an n-set X then, for n>=8, a(n-8) is the number of 3-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 16 2007

Links

Crossrefs

Cf. A000292, A096956.
Cf. other columns: A096958 (m = 4), A096959 (m = 5), A097297 (m = 6), A097298 (m = 7), A097299 (m = 8), A097300 (m = 9).

Programs

  • Magma
    I:=[6,19,40,70]; [n le 4 select I[n] else 4*Self(n-1)- 6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Apr 19 2017
  • Mathematica
    CoefficientList[Series[(6 - 5*x)/(1 - x)^4, {x, 0, 40}], x] (* Wesley Ivan Hurt, Apr 18 2017 *)
LinearRecurrence[{4, -6, 4, -1}, {6, 19, 40, 70}, 50] (* Vincenzo Librandi, Apr 19 2017 *)

Formula

a(n) = A096956(n+3, 3) = 6*b(n) - 5*b(n-1) = (n+18)*binomial(n+2, 2)/3, with b(n) = A000292(n) = binomial(n+3, 3).
G.f.: (6-5*x)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Apr 19 2017
E.g.f.: exp(x)*(36 + 78*x + 24*x^2 + x^3)/6. - Stefano Spezia, May 02 2025

A097297 Seventh column (m=6) of (1,6)-Pascal triangle A096956.

Original entry on oeis.org

6, 37, 133, 364, 840, 1722, 3234, 5676, 9438, 15015, 23023, 34216, 49504, 69972, 96900, 131784, 176358, 232617, 302841, 389620, 495880, 624910, 780390, 966420, 1187550, 1448811, 1755747, 2114448, 2531584, 3014440, 3570952, 4209744
Offset: 0

Views

Author

Wolfdieter Lang, Aug 13 2004

Keywords

Links

Crossrefs

Cf. A000579, A096956.
Cf. other columns: A096957 (m = 3), A096958 (m = 4), A096959 (m = 5), A097298 (m = 7), A097299 (m = 8), A097300 (m = 9).

Programs

Formula

a(n) = A096956(n+6, 6) = 6*b(n) - 5*b(n-1) = (n+36)*binomial(n+5, 5)/6, with b(n) = A000579(n+6) = binomial(n+6, 6).
G.f.: (6-5*x)/(1-x)^7.

A096959 Sixth column (m=5) of (1,6)-Pascal triangle A096956.

Original entry on oeis.org

6, 31, 96, 231, 476, 882, 1512, 2442, 3762, 5577, 8008, 11193, 15288, 20468, 26928, 34884, 44574, 56259, 70224, 86779, 106260, 129030, 155480, 186030, 221130, 261261, 306936, 358701, 417136, 482856, 556512, 638792, 730422, 832167, 944832
Offset: 0

Views

Author

Wolfdieter Lang, Aug 13 2004

Keywords

Links

Crossrefs

Cf. A096958 (fifth column), A097297 (seventh column).

Programs

  • Magma
    [(n+30)*Binomial(n+4, 4)/5: n in [0..30]]; // G. C. Greubel, Nov 24 2017
  • Mathematica
    Table[(n + 30)*Binomial[n + 4, 4]/5, {n, 0, 50}] (* G. C. Greubel, Nov 24 2017 *)
  • PARI
    for(n=0,30, print1((n+30)*binomial(n+4, 4)/5, ", ")) \\ G. C. Greubel, Nov 24 2017

Formula

a(n) = A096956(n+5, 5).
a(n) = 6*b(n) - 5*b(n-1), with b(n) = A000389(n+5) = binomial(n+5, 5).
a(n) = (n+30)*binomial(n+4, 4)/5.
G.f.: (6-5*x)/(1-x)^6.
E.g.f.: x*(720 + 1140*x + 420*x^2 + 45*x^3 + x^4)*exp(x)/120. - G. C. Greubel, Nov 24 2017

A097300 Tenth column (m=9) of (1,6)-Pascal triangle A096956.

Original entry on oeis.org

6, 55, 280, 1045, 3190, 8437, 20020, 43615, 88660, 170170, 311168, 545870, 923780, 1514870, 2416040, 3759074, 5720330, 8532425, 12498200, 18007275, 25555530, 35767875, 49424700, 67492425, 91158600, 121872036, 161388480, 211822380
Offset: 0

Views

Author

Wolfdieter Lang, Aug 13 2004

Keywords

Links

Crossrefs

Cf. A000582, A096956.
Cf. other columns: A096957 (m = 3), A096958 (m = 4), A096959 (m = 5), A097297 (m = 6), A097298 (m = 7), A097299 (m = 8).

Programs

Formula

a(n) = A096956(n+9, 9) = 6*b(n) - 5*b(n-1) = (n+54)*binomial(n+8, 8)/9, with b(n) = A000582(n+9) = binomial(n+9, 9).
G.f.: (6-5*x)/(1-x)^10.

A097298 Eighth column (m=7) of (1,6)-Pascal triangle A096956.

Original entry on oeis.org

6, 43, 176, 540, 1380, 3102, 6336, 12012, 21450, 36465, 59488, 93704, 143208, 213180, 310080, 441864, 618222, 850839, 1153680, 1543300, 2039180, 2664090, 3444480, 4410900, 5598450, 7047261, 8803008, 10917456, 13449040, 16463480
Offset: 0

Views

Author

Wolfdieter Lang, Aug 13 2004

Keywords

Links

Crossrefs

Cf. A000580, A096956.
Cf. other columns: A096957 (m = 3), A096958 (m = 4), A096959 (m = 5), A097297 (m = 6), A097299 (m = 8), A097300 (m = 9).

Programs

Formula

a(n) = A096956(n+7, 7) = 6*b(n) - 5*b(n-1) = (n+42)*binomial(n+6, 6)/7, with b(n) = A000580(n+7) = binomial(n+7, 7).
G.f.: (6-5*x)/(1-x)^8.

A097299 Ninth column (m=8) of (1,6)-Pascal triangle A096956.

Original entry on oeis.org

6, 49, 225, 765, 2145, 5247, 11583, 23595, 45045, 81510, 140998, 234702, 377910, 591090, 901170, 1343034, 1961256, 2812095, 3965775, 5509075, 7548255, 10212345, 13656825, 18067725, 23666175, 30713436, 39516444, 50433900, 63882940
Offset: 0

Views

Author

Wolfdieter Lang, Aug 13 2004

Keywords

Links

Crossrefs

Cf. A000581, A096956.
Cf. other columns: A096957 (m = 3), A096958 (m = 4), A096959 (m = 5), A097297 (m = 6), A097298 (m = 7), A097300 (m = 9).

Programs

Formula

a(n) = A096956(n+8, 8) = 6*b(n) - 5*b(n-1) = (n+48)*binomial(n+7, 7)/8, with b(n) = A000581(n+8) = binomial(n+8, 8).
G.f.: (6-5*x)/(1-x)^9.
Showing 1-6 of 6 results.

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