A190088 -id:A190088 - cp's OEIS frontend

A190089 Row sums of the triangular matrix A190088.

1, 4, 21, 114, 616, 3329, 17991, 97229, 525456, 2839729, 15346786, 82938844, 448227521, 2422362079, 13091204281, 70748973084, 382349636061, 2066337330754, 11167134898976, 60350698792449, 326154101090951, 1762639037938629, 9525854090667496, 51480702630305689, 278217860370802066

Offset: 0

Emanuele Munarini, May 04 2011

G. C. Greubel, Table of n, a(n) for n = 0..1350 (terms 0..136 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (5,2,1).

Cf. A190088, A190090.

[(&+[Binomial(3*n-k+1,3*n-3*k+1): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Mar 04 2018

Table[Sum[Binomial[3n - k + 1, 3n - 3k + 1], {k, 0, n}], {n, 0, 12}]
LinearRecurrence[{5,2,1},{1,4,21},30] (* Harvey P. Dale, Sep 18 2013 *)

makelist(sum(binomial(3*n-k+1,3*n-3*k+1),k,0,n),n,0,24);

Vec((1-x-x^2)/(1-5*x-2*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Jun 30 2011

a(n) = Sum_{k=0..n} binomial(3*n-k+1,3*n-3*k+1).
G.f.: (1-x-x^2)/(1-5*x-2*x^2-x^3).
a(n) = 5*a(n-1)+2*a(n-2)+a(n-3) and a(0)=1, a(1)=4, a(2)=21. - Harvey P. Dale, Sep 18 2013

A190090 Diagonal sums of the triangular matrix A190088.

Original entry on oeis.org

1, 1, 4, 16, 42, 137, 443, 1365, 4316, 13625, 42785, 134758, 424331, 1335378, 4203927, 13233947, 41657808, 131135696, 412803240, 1299458257, 4090567673, 12876698159, 40534529294, 127598621869, 401667591501, 1264408966284, 3980231826575, 12529367967276, 39441185140197

Offset: 0

Emanuele Munarini, May 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..201
Index entries for linear recurrences with constant coefficients, signature (2,2,6,-3,0,1).

Cf. A190088, A190089.

[(&+[Binomial(3*n-4*k+1,3*n-6*k+1): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Mar 04 2018

Table[Sum[Binomial[3n - 4k + 1, 3n - 6k + 1], {k, 0, n/2}], {n, 0, 26}]
LinearRecurrence[{2,2,6,-3,0,1},{1,1,4,16,42,137},27] (* Harvey P. Dale, Jul 04 2011 *)

makelist(sum(binomial(3*n-4*k+1,3*n-6*k+1),k,0,n/2),n,0,12);

Vec((1-x-x^4)/(1-2*x-2*x^2-6*x^3+3*x^4-x^6)+O(x^29)) \\ Charles R Greathouse IV, Jun 30 2011

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-4*k+1,3*n-6*k+1).
G.f.: (1-x-x^4)/(1-2*x-2*x^2-6*x^3+3*x^4-x^6).
a(n) = 2*a(n-1)+ 2*a(n-2)+ 6*a(n-3)-3*a(n-4)+a(n-6), and a(0)=1, a(1)=1, a(2)=4, a(3)=16, a(4)=42, a(5)=137, . - Harvey P. Dale, Jul 04 2011

A190152 Triangle of binomial coefficients binomial(3n-k,3n-3*k).

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 28, 35, 1, 1, 55, 210, 84, 1, 1, 91, 715, 924, 165, 1, 1, 136, 1820, 5005, 3003, 286, 1, 1, 190, 3876, 18564, 24310, 8008, 455, 1, 1, 253, 7315, 54264, 125970, 92378, 18564, 680, 1, 1, 325, 12650, 134596, 490314, 646646, 293930, 38760, 969, 1

Offset: 0

Emanuele Munarini, May 05 2011

From R. J. Mathar, Mar 15 2013: (Start)
The matrix inverse starts
  1;
  -1,1;
  9,-10,1;
  -288,322,-35,1;
  22356,-25003,2730,-84,1;
  -3428973,3835026,-418825,12936,-165,1;
  914976405,-1023326973,111759115,-3452449,44187,-286,1;
  ... (End)

			Triangle begins:
  1
  1, 1
  1, 10, 1
  1, 28, 35, 1
  1, 55, 210, 84, 1
  1, 91, 715, 924, 165, 1
  1, 136, 1820, 5005, 3003, 286, 1
  1, 190, 3876, 18564, 24310, 8008, 455, 1
  1, 253, 7315, 54264, 125970, 92378, 18564, 680, 1
  ...

Nathaniel Johnston, Rows n = 0..100, flattened

Cf. A000447 (first subdiagonal), A053135 (second subdiagonal), A060544 (second column), A190088, A190153 (row sums), A190154 (diagonal sums).

Flatten[Table[Binomial[3n - k, 3n - 3k], {n, 0, 9}, {k, 0, n}]]

create_list(binomial(3*n-k,3*n-3*k),n,0,9,k,0,n);

for(n=0,10, for(k=0,n, print1(binomial(3*n-k, 3*(n-k)), ", "))) \\ G. C. Greubel, Dec 29 2017

cp's OEIS Frontend

A190089 Row sums of the triangular matrix A190088.

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A190090 Diagonal sums of the triangular matrix A190088.

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A190152 Triangle of binomial coefficients binomial(3n-k,3n-3*k).

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cp's OEIS Frontend

A190089 Row sums of the triangular matrix A190088.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A190090 Diagonal sums of the triangular matrix A190088.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A190152 Triangle of binomial coefficients binomial(3*n-k,3*n-3*k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

A190152 Triangle of binomial coefficients binomial(3n-k,3n-3*k).