A036219 -id:A036219 - cp's OEIS frontend

A172362 a(n) = binomial(n+10, 10)*3^n.

1, 33, 594, 7722, 81081, 729729, 5837832, 42532776, 287096238, 1818276174, 10909657044, 62482581252, 343654196886, 1824010737318, 9380626649064, 46903133245320, 228652774570935, 1089463220014455, 5084161693400790

Offset: 0

Zerinvary Lajos, Feb 01 2010

With a different offset, number of n-permutations (n>=10) of 4 objects: u, v, z, x with repetition allowed, containing exactly ten, (10) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (33,-495,4455,-26730,112266,-336798,721710,-1082565,1082565,-649539,177147).

Cf. A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223.

[3^n*Binomial(n+10, 10): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Maple
```
seq(binomial(n+10, 10)*3^n, n=0..30);
```

Table[Binomial[n + 10, 10]*3^n, {n, 0, 20}]

G.f.: 1/(1-3*x)^11. - Vincenzo Librandi, Oct 15 2011
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 261617/42 - 15360*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 7864320*log(4/3) - 47510881/21. (End)

A099097 Riordan array (1, 3+x).

Original entry on oeis.org

1, 0, 3, 0, 1, 9, 0, 0, 6, 27, 0, 0, 1, 27, 81, 0, 0, 0, 9, 108, 243, 0, 0, 0, 1, 54, 405, 729, 0, 0, 0, 0, 12, 270, 1458, 2187, 0, 0, 0, 0, 1, 90, 1215, 5103, 6561, 0, 0, 0, 0, 0, 15, 540, 5103, 17496, 19683, 0, 0, 0, 0, 0, 1, 135, 2835, 20412, 59049, 59049, 0, 0, 0, 0, 0, 0, 18, 945, 13608, 78732, 196830, 177147

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A006190(n+1). Diagonal sums are A052931. The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1/3, -1/3, 0, 0, 0, 0, 0, ...] DELTA [3, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008

			Triangle begins:
  1;
  0, 3;
  0, 1, 9;
  0, 0, 6, 27;
  0, 0, 1, 27,  81;
  0, 0, 0,  9, 108, 243;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A027465.

Diagonals are of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), A036222 (m=8), A036223 (m=9), A172362 (m=10).

Table[3^(2*k-n)*Binomial[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 19 2021 *)

flatten([[3^(2*k-n)*binomial(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 19 2021

Triangle: T(n, k) = binomial(k, n-k)*3^k*(1/3)^(n-k).
G.f. of column k: (3*x + x^2)^k.
G.f.: 1/(1 - 3*y*x - y*x^2). - Philippe Deléham, Nov 21 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A006190(n+1), A135030(n+1), A181353(n+1) for x = 0,1,2,3 respectively. - Philippe Deléham, Nov 21 2011

A140404 a(n) = binomial(n+5, 5)*7^n.

Original entry on oeis.org

1, 42, 1029, 19208, 302526, 4235364, 54353838, 652246056, 7419298887, 80787921214, 848273172747, 8636963213424, 85649885199788, 830145041167176, 7886377891088172, 73606193650156272, 676256904160810749, 6126091955339109138, 54794489156088698401, 484498640959100070072

Offset: 0

Zerinvary Lajos, Jun 16 2008

With a different offset, number of n-permutations of 8 objects:r,s,t,u,v,z,x,y with repetition allowed, containing exactly five (5) u's. Example: a(1)=42 because we have
uuuuur, uuuuru, uuuruu, uuruuu, uruuuu, ruuuuu
uuuuus, uuuusu, uuusuu, uusuuu, usuuuu, suuuuu,
uuuuut, uuuutu, uuutuu, uutuuu, utuuuu, tuuuuu,
uuuuuv, uuuuvu, uuuvuu, uuvuuu, uvuuuu, vuuuuu,
uuuuuz, uuuuzu, uuuzuu, uuzuuu, uzuuuu, zuuuuu,
uuuuux, uuuuxu, uuuxuu, uuxuuu, uxuuuu, xuuuuu,
uuuuuy, uuuuyu, uuuyuu, uuyuuu, uyuuuu, yuuuuu.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (42,-735,6860,-36015,100842,-117649).

Cf. A000389, A036084, A036219, A045543, A050982, A054849.

[7^n* Binomial(n+5, 5): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Maple
```
seq(binomial(n+5,5)*7^n,n=0..17);
```

Table[Binomial[n+5,5]7^n,{n,0,20}] (* or *) LinearRecurrence[ {42,-735,6860,-36015,100842,-117649},{1,42,1029,19208,302526,4235364},21] (* Harvey P. Dale, Sep 08 2011 *)

a(n)=binomial(n+5,5)*7^n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 1/(1-7*x)^6. - Zerinvary Lajos, Aug 06 2008
a(n) = 42*a(n-1) - 735*a(n-2) + 6860*a(n-3) - 36015*a(n-4) + 100842*a(n-5) - 117649*a(n-6). - Harvey P. Dale, Sep 08 2011
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 45360*log(7/6) - 27965/4.
Sum_{n>=0} (-1)^n/a(n) = 143360*log(8/7) - 229705/12. (End)

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A172362 a(n) = binomial(n+10, 10)*3^n.

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A099097 Riordan array (1, 3+x).

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A140404 a(n) = binomial(n+5, 5)*7^n.

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