A090285 -id:A090285 - cp's OEIS frontend

A090317 Row sums of triangle in A090285.

1, 2, 7, 28, 118, 510, 2235, 9876, 43870, 195556, 873814, 3911168, 17527904, 78622982, 352911939, 1584927828, 7120769526, 32002212252, 143859840114, 646819996008, 2908670252676, 13081556909292, 58839348572574, 264674150692488, 1190649451348908, 5356483791828840, 24098774900561500

Offset: 0

Philippe Deléham, Jan 25 2004

Apply the inverse of the Riordan array (1/(1-x^2),x/(1+x)^2) to 2^n. - Paul Barry, Mar 13 2009

Hankel transform is A079935. - Paul Barry, Mar 13 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A001700, A090285.

Table[SeriesCoefficient[(1-x^2*((1-Sqrt[1-4*x])/(2*x))^4)/(1-2*x*((1-Sqrt[1-4*x])/(2*x))^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

a(n):=if n=0 then 1 else 4*binomial(2*n-1,n)/(n+1)+3*sum(((k+1)*2^(k)*binomial(2*n-1,n-k-1))/(n+k+1),k,1,n-1); /* Vladimir Kruchinin, Feb 21 2019 */

x='x+O('x^66); Vec((1-x^2*((1-sqrt(1-4*x))/(2*x))^4)/(1-2*x*((1-sqrt(1-4*x))/(2*x))^2)) \\ Joerg Arndt, May 11 2013

a(n+1) = A000108(n+1) + Sum_{k=0..n} a(n-k)*A001700(k); a(0) = 1.
G.f.: (1-x^2*c(x)^4)/(1-2x*c(x)^2), where c(x) is the g.f. of the Catalan numbers A000108. - Paul Barry, Mar 13 2009
Recurrence: 2*(n+1)*(n+3)*a(n) = (17*n^2+56*n-21)*a(n-1) - 18*(n+4)*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 9^n/2^(n+2). - Vaclav Kotesovec, Oct 14 2012
a(n) = 4*C(2*n-1,n)/(n+1)+3*Sum_{k=1..n-1}(k+1)*2^k*C(2*n-1,n-k-1)/(n+k+1), n>0, a(0)=1. - Vladimir Kruchinin, Feb 21 2019

Term 15 corrected by Paul Barry, Mar 13 2009

A090294 a(n) = K_3(n) = Sum_{k>=0} A090285(3,k)2^kbinomial(n,k). a(n) = (4n^3+30n^2+56*n+15)/3.

Original entry on oeis.org

5, 35, 93, 187, 325, 515, 765, 1083, 1477, 1955, 2525, 3195, 3973, 4867, 5885, 7035, 8325, 9763, 11357, 13115, 15045, 17155, 19453, 21947, 24645, 27555, 30685, 34043, 37637, 41475, 45565, 49915, 54533, 59427, 64605, 70075, 75845, 81923, 88317

Offset: 0

Philippe Deléham, Jan 25 2004

Values of polynomial K_3 related to A090285.

Cf. A090285.

A090294:=n->(4*n^3+30*n^2+56*n+15)/3; seq(A090294(n), n=0..100); # Wesley Ivan Hurt, Dec 19 2013

Table[(4 n^3 + 30 n^2 + 56 n + 15)/3, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 19 2013 *)

O.g.f.: 8/(-1+x)^4+5/(-1+x)-2/(-1+x)^2-4/(-1+x)^3 . - R. J. Mathar, Feb 26 2008

A090297 a(n) = K_5(n) = Sum_{k>=0} A090285(5,k)2^kbinomial(n,k). a(n) = 2(2n^5+45n^4+360n^3+1215n^2+1528n+315)/15.

Original entry on oeis.org

42, 462, 1586, 3958, 8330, 15694, 27314, 44758, 69930, 105102, 152946, 216566, 299530, 405902, 540274, 707798, 914218, 1165902, 1469874, 1833846, 2266250, 2776270, 3373874, 4069846, 4875818, 5804302, 6868722, 8083446, 9463818

Offset: 0

Philippe Deléham, Jan 25 2004

Values of polynomial K_5 related to A090285.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A090285.

[2*(2*n^5 + 45*n^4 + 360*n^3 + 1215*n^2 + 1528*n + 315)/15: n in [0..30]]; // Vincenzo Librandi, Sep 18 2012

Table[(2*(2*n^5 + 45*n^4 + 360*n^3 + 1215*n^2 + 1528*n + 315)/15),{n, 0, 50}] (* Vincenzo Librandi, Sep 18 2012  *)
LinearRecurrence[{6,-15,20,-15,6,-1},{42,462,1586,3958,8330,15694},30] (* Harvey P. Dale, Apr 17 2020 *)

G.f.: (42+210*x-556*x^2+532*x^3-238*x^4+42*x^5)/(1-x)^6. [Colin Barker, Sep 18 2012]

Corrected by T. D. Noe, Nov 09 2006

A090296 a(n) = K_4(n) = Sum_{k>=0} A090285(4,k)2^kbinomial(n,k). a(n) = 2(n^4+14n^3+62n^2+91n+21)/3.

Original entry on oeis.org

14, 126, 386, 874, 1686, 2934, 4746, 7266, 10654, 15086, 20754, 27866, 36646, 47334, 60186, 75474, 93486, 114526, 138914, 166986, 199094, 235606, 276906, 323394, 375486, 433614, 498226, 569786, 648774, 735686, 831034, 935346, 1049166

Offset: 0

Philippe Deléham, Jan 25 2004

Values of polynomial K_4 related to A090285.

Cf. A090285.

A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 5, 1, 14, 35, 22, 7, 1, 42, 126, 93, 38, 9, 1, 132, 462, 386, 187, 58, 11, 1, 429, 1716, 1586, 874, 325, 82, 13, 1, 1430, 6435, 6476, 3958, 1686, 515, 110, 15, 1, 4862, 24310, 26333, 17548, 8330, 2934, 765, 142, 17, 1

Offset: 0

Philippe Deléham, Jan 25 2004

Read as a number triangle, this is the Riordan array (c(x),x/sqrt(1-4x)) where c(x) is the g.f. of A000108. - Paul Barry, May 16 2005

			row n=0 : 1, 1, 2, 5, 14, 42, 132, 429, ... see A000108.
row n=1 : 1, 3, 10, 35, 126, 462, 1716, 6435, ... see A001700.
row n=2 : 1, 5, 22, 93, 386, 1586, 6476, ... see A000346.
row n=3 : 1, 7, 38, 187, 874, 3958, 17548, ... see A000531.
row n=4 : 1, 9, 58, 325, 1686, 8330, 39796, ... see A018218.

Other rows : A029887, A042941, A045724, A042985, A045492. Columns : A000012, A005408. Row n is the convolution of the row (n-j) with A000984, A000302, A002457, A002697 (first term omitted), A002802, A038845, A020918, A038846, A020920 for j=1, 2, ..9 respectively.

T(n, k) = K_k(n)= Sum_{j>=0} A090285(k, j)*2^j*binomial(n, j). T(n, 1) = 2*n+1. T(n, 2) = 2*A028387(n).

Corrected by Alford Arnold, Oct 18 2006

A090288 a(n) = 2n^2 + 6n + 2.

Original entry on oeis.org

2, 10, 22, 38, 58, 82, 110, 142, 178, 218, 262, 310, 362, 418, 478, 542, 610, 682, 758, 838, 922, 1010, 1102, 1198, 1298, 1402, 1510, 1622, 1738, 1858, 1982, 2110, 2242, 2378, 2518, 2662, 2810, 2962, 3118, 3278, 3442, 3610, 3782, 3958, 4138, 4322, 4510

Offset: 0

Philippe Deléham, Jan 25 2004

Values of polynomial K_2 related to A090285: a(n) = K_2(n) = Sum_{k>=0} A090285(2,k)*2^k*binomial(n,k).
Numbers k such that 2*k+5 is a square. - Vincenzo Librandi, Oct 10 2013
a(n) is the area of a triangle with vertices at (b(n-2),b(n-1)), (b(n),b(n+1)), and (b(n+2),B(n+3)) for b(k)=A000292(k) with n>1. - J. M. Bergot, Mar 23 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028387, A090285.

List([0..50], n-> 2*(1+3*n+n^2)); # G. C. Greubel, May 31 2019

[2*(1+3*n+n^2): n in [0..50]]; // G. C. Greubel, May 31 2019

Table[2*(n^2 +3*n +1), {n, 0, 50}] (* Vincenzo Librandi, Oct 10 2013 *)
LinearRecurrence[{3,-3,1},{2,10,22},50] (* Harvey P. Dale, May 04 2017 *)

a(n)=2*n^2+6*n+2 \\ Charles R Greathouse IV, Sep 24 2015

[2*(1+3*n+n^2) for n in (0..50)] # G. C. Greubel, May 31 2019

a(n) = 2*A028387(n).
G.f.: 2*(1 +2*x -x^2)/(1-x)^3. - R. J. Mathar, Apr 02 2008
E.g.f.: 2*(1 +4*x +x^2)*exp(x). - G. C. Greubel, Jul 13 2017
Sum_{n>=0} 1/a(n) = 1/2 + Pi*tan(sqrt(5)*Pi/2)/(2*sqrt(5)). - Amiram Eldar, Dec 23 2022

Corrected by T. D. Noe, Nov 12 2006

A236471 Riordan array ((1-x)/(1-2x), x(1-x)/(1-2x)^2).

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 4, 13, 7, 1, 8, 38, 33, 10, 1, 16, 104, 129, 62, 13, 1, 32, 272, 450, 304, 100, 16, 1, 64, 688, 1452, 1289, 590, 147, 19, 1, 128, 1696, 4424, 4942, 2945, 1014, 203, 22, 1, 256, 4096, 12896, 17584, 13073, 5823, 1603, 268, 25, 1, 512, 9728

Offset: 0

Philippe Deléham, Jan 26 2014

Row sums are A052936(n).
Diagonal sums are A121449(n).
The triangle T'(n,k) = T(n,k)*(-1)^(n+k) is the inverse of the Riordan array in A090285.

			Triangle begins:
1;
1, 1;
2, 4, 1;
4, 13, 7, 1;
8, 38, 33, 10, 1;
16, 104, 129, 62, 13, 1;
32, 272, 450, 304, 100, 16, 1;
64, 688, 1452, 1289, 590, 147, 19, 1;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A011782, A049611, A000012, A016777, A062708.

CoefficientList[CoefficientList[Series[(2*x^2-3*x+1)/((x^2-x)*y +4*x^2 - 4*x+1), {x,0,20}, {y,0,20}], x], y]//Flatten (* G. C. Greubel, Apr 19 2018 *)

T(n,k):=sum(binomial(m+k,2*k)*binomial(n-1,n-m),m,0,n); /* Vladimir Kruchinin, Apr 21 2015 */

for(n=0,20, for(k=0,n, print1(sum(m=0, n, binomial(m+k,2*k)* binomial(n-1,n-m)), ", "))) \\ G. C. Greubel, Apr 19 2018

T(n,0) = A011782(n), T(n,1) = A049611(n), T(n,n) = A000012(n) = 1, T(n+1,n) = A016777(n), T(n+2,n) = A062708(n+1).
G.f.: (2*x^2-3*x+1)/((x^2-x)*y+4*x^2-4*x+1). - Vladimir Kruchinin, Apr 21 2015
T(n,k) = Sum_{m=0..n} C(m+k,2*k)*C(n-1,n-m). - Vladimir Kruchinin, Apr 21 2015

cp's OEIS Frontend

A090317 Row sums of triangle in A090285.

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A090294 a(n) = K_3(n) = Sum_{k>=0} A090285(3,k)*2^k*binomial(n,k). a(n) = (4*n^3+30*n^2+56*n+15)/3.

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A090297 a(n) = K_5(n) = Sum_{k>=0} A090285(5,k)*2^k*binomial(n,k). a(n) = 2*(2*n^5+45*n^4+360*n^3+1215*n^2+1528*n+315)/15.

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A090296 a(n) = K_4(n) = Sum_{k>=0} A090285(4,k)*2^k*binomial(n,k). a(n) = 2*(n^4+14*n^3+62*n^2+91*n+21)/3.

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A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

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A090288 a(n) = 2*n^2 + 6*n + 2.

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A236471 Riordan array ((1-x)/(1-2*x), x(1-x)/(1-2*x)^2).

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A090294 a(n) = K_3(n) = Sum_{k>=0} A090285(3,k)2^kbinomial(n,k). a(n) = (4n^3+30n^2+56*n+15)/3.

A090297 a(n) = K_5(n) = Sum_{k>=0} A090285(5,k)2^kbinomial(n,k). a(n) = 2(2n^5+45n^4+360n^3+1215n^2+1528n+315)/15.

A090296 a(n) = K_4(n) = Sum_{k>=0} A090285(4,k)2^kbinomial(n,k). a(n) = 2(n^4+14n^3+62n^2+91n+21)/3.

A090288 a(n) = 2n^2 + 6n + 2.

A236471 Riordan array ((1-x)/(1-2x), x(1-x)/(1-2x)^2).