A165241 -id:A165241 - cp's OEIS frontend

A202481 Column k = 3 of triangular array in A165241.

1, 10, 55, 231, 833, 2720, 8280, 23920, 66352, 178176, 465920, 1191680, 2991360, 7389184, 17999872, 43315200, 103116800, 243138560, 568393728, 1318518784, 3037265920, 6952058880, 15820390400, 35809918976, 80659611648

Offset: 0

Philippe Deléham, Dec 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A011782, A084858, A165241, A202480

LinearRecurrence[{8,-24,32,-16},{1,10,55,231,833},30] (* Harvey P. Dale, Dec 04 2018 *)

G.f.: (1+2x-x^2-x^3+x^4)/(1-2x)^4.
a(n) = A165241(n+3,3).
a(n) = 2^n*(n+3)*(3*n+2)*(3*n+5)/32 for n>0. - Bruno Berselli, Dec 21 2011

A202493 Column k = 4 of triangular array in A165241.

Original entry on oeis.org

1, 15, 105, 532, 2241, 8361, 28610, 91740, 279624, 818272, 2315712, 6372480, 17123840, 45082368, 116596224, 296879104, 745543680, 1849344000, 4536958976, 11020075008, 26526547968, 63329075200, 150057123840

Offset: 0

Philippe Deléham, Dec 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Cf. A011782, A084858, A165241, A202480, A202481.

a(n) = A165241(n+4,4).
G.f.: (1+5x-5x^2+2x^3+x^4-x^5)/(1-2x)^5.
a(n) = 2^n*(n+4)*(27*n^3+126*n^2+169*n+62)/256 for n>0. - Bruno Berselli, Dec 21 2011

a(12) added by Bruno Berselli, Dec 21 2011

A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.

Original entry on oeis.org

1, 3, 2, 6, 9, 4, 10, 25, 24, 8, 15, 55, 85, 60, 16, 21, 105, 231, 258, 144, 32, 28, 182, 532, 833, 728, 336, 64, 36, 294, 1092, 2241, 2720, 1952, 768, 128, 45, 450, 2058, 5301, 8361, 8280, 5040, 1728, 256, 55, 660, 3630, 11385, 22363, 28610, 23920

Offset: 1

Clark Kimberling, Mar 25 2012

Column 1: triangular numbers, A000217
Coefficient of v(n,x):  2^(n-1)
Row sums:  A035344
Alternating row sums: 1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
Appears to be the reversed row polynomials of A165241 with the unit diagonal removed. If so, the o.g.f. is [1-(1+y)x]/[1-2(1+y)x+(1+y)x^2] - 1/(1-x) and the triangular matrix here may be formed by adding each column of the matrix of A056242, presented in the example section with the additional zeros, to its subsequent column with the first row ignored. - Tom Copeland, Jan 09 2017

			First five rows:
1
3....2
6....9....4
10...25...24...8
15...55...85...60...16
First three polynomials v(n,x): 1, 3 + 2x, 6 + 9x +4x^2

Cf. A210753, A208510.

Cf. A056242, A165241.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210754 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)

u(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A202480 Riordan array (1/(1-x), x(2x-1)/(1-x)^2).

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, 0, 1, -1, 1, 2, -1, -1, 1, 1, 5, -5, 2, 1, -1, 1, 9, -10, 8, -3, -1, 1, 1, 14, -14, 14, -11, 4, 1, -1, 1, 20, -14, 14, -17, 14, -5, -1, 1, 1, 27, -6, 0, -9, 19, -17, 6, 1, -1

Offset: 0

Philippe Deléham, Dec 20 2011

Row sums are Fibonacci(n-1) = A000045(n-1).
Diagonal sums are A078003(n).
(Sum_{j, 0<=j<=k} T(k,j))/(1-2x)^k gives g.f. of column A165241(n+k-1,k-1) in triangular array in A165241.

			Triangle begins :
1
1, -1
1, -1, 1
1, 0, 1, -1
1, 2, -1, -1, 1
1, 5, -5, 2, 1, -1
1, 9, -10, 8, -3, -1, 1
1, 14, -14, 14, -11, 4, 1, -1
(1+x^2-x^3)/(1-2x)^3 is the g.f of column A165241(n+2,2) := 1, 6, 25, 85, 258, 728, 1952, 5040, ...

Cf. A000012, A080956, A000045, A078003, A124341, A165241

T(n,k) = 2*T(n-1,k) + 2*T(n-2,k-1) - T(n-1,k-1) - T(n-2,k).

T(n,k) = (-1)^n*A124341(n,k).

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A202481 Column k = 3 of triangular array in A165241.

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A202493 Column k = 4 of triangular array in A165241.

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A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.

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A202480 Riordan array (1/(1-x), x(2x-1)/(1-x)^2).

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