A097298 -id:A097298 - cp's OEIS frontend

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

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

Wolfdieter Lang, Aug 13 2004

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

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A096956.

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

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

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 *)

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

Wolfdieter Lang, Aug 13 2004

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000579, A096956.

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

A097297[n_] := (n + 36)*Binomial[n + 5, 5]/6;
Array[A097297, 50, 0] (* Paolo Xausa, May 02 2025 *)

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.

A096958 Fifth column (m=4) of (1,6)-Pascal triangle A096956.

Original entry on oeis.org

6, 25, 65, 135, 245, 406, 630, 930, 1320, 1815, 2431, 3185, 4095, 5180, 6460, 7956, 9690, 11685, 13965, 16555, 19481, 22770, 26450, 30550, 35100, 40131, 45675, 51765, 58435, 65720, 73656, 82280, 91630, 101745, 112665, 124431, 137085, 150670

Offset: 0

Wolfdieter Lang, Aug 13 2004

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

Cf. A000332, A096956.

Cf. other columns: A096957 (m = 3), A096959 (m = 5), A097297 (m = 6), A097298 (m = 7), A097299 (m = 8), A097300 (m = 9).

[(n+24)*Binomial(n+3, 3) div 4: n in [0..40]]; // Vincenzo Librandi, Oct 01 2013

Table[(n + 24) Binomial[n+3, 3]/4, {n, 0, 50}] (* Vincenzo Librandi, Oct 01 2013 *)

a(n) = A096956(n+4, 4) = 6*b(n) - 5*b(n-1) = (n+24)*binomial(n+3, 3)/4, with b(n) = A000332(n) = binomial(n+4, 4).
G.f.: (6-5*x)/(1-x)^5.
a(n) = sum_{k=1..n+1} ( sum_{i=1..k} i*(n-k+7) ). - Wesley Ivan Hurt, Sep 26 2013

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

Wolfdieter Lang, Aug 13 2004

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000582, A096956.

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

A097300[n_] := (n + 54)*Binomial[n + 8, 8]/9;
Array[A097300, 50, 0] (* Paolo Xausa, May 02 2025 *)

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.

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

Wolfdieter Lang, Aug 13 2004

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000581, A096956.

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

A097299[n_] := (n + 48)*Binomial[n + 7, 7]/8;
Array[A097299, 50, 0] (* Paolo Xausa, May 02 2025 *)

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.

cp's OEIS Frontend

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

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

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A096958 Fifth column (m=4) of (1,6)-Pascal triangle A096956.

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

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

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