A072024 -id:A072024 - cp's OEIS frontend

A081297 Array T(k,n), read by antidiagonals: T(k,n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 13, 13, 11, 1, 1, 1, 21, 25, 55, 21, 1, 1, 1, 31, 41, 181, 133, 43, 1, 1, 1, 43, 61, 461, 481, 463, 85, 1, 1, 1, 57, 85, 991, 1281, 2653, 1261, 171, 1, 1, 1, 73, 113, 1891, 2821, 10501, 8425, 4039, 341, 1, 1, 1, 91, 145, 3305

Offset: 0

Paul Barry, Mar 17 2003

Square array of solutions of a family of recurrences.
Rows of the array give solutions to the recurrences a(n)=a(n-1)+k(k-1)a(n-2), a(0)=a(1)=1.
Subarray of array in A072024. - Philippe Deléham, Nov 24 2013

			Rows begin
  1, 1,  1,  1,   1,    1, ...
  1, 1,  3,  5,  11,   21, ...
  1, 1,  7, 13,  55,  133, ...
  1, 1, 13, 25, 181,  481, ...
  1, 1, 21, 41, 461, 1281, ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Cf. A059259, A072024.
Rows include A001045, A015441, A053404, A053428, A053430.
Columns include A002061, A001844, A072025.
Diagonals include A081298, A081299, A081300, A081301, A081302.

T[n_, k_]:=((n + 1)^(k + 1) - (-n)^(k + 1)) / (2n + 1); Flatten[Table[T[n - k, k], {n, 0, 10}, {k, 0, n}]] (* Indranil Ghosh, Mar 27 2017 *)

for(k=0, 10, for(n=0, 9, print1(((k+1)^(n+1)-(-k)^(n+1))/(2*k+1), ", "); ); print(); ) \\ Andrew Howroyd, Mar 26 2017

def T(n, k): return ((n + 1)**(k + 1) - (-n)**(k + 1)) // (2*n + 1)
for n in range(11):
    print([T(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Mar 27 2017

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

Rows of the array have g.f. 1/((1+kx)(1-(k+1)x)).

Name clarified by Andrew Howroyd, Mar 27 2017

A080242 Table of coefficients of polynomials P(n,x) defined by the relation P(n,x) = (1+x)*P(n-1,x) + (-x)^(n+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 1, 4, 7, 6, 3, 1, 5, 11, 13, 9, 3, 1, 1, 6, 16, 24, 22, 12, 4, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, 10, 46, 128, 239, 314, 296, 200, 95

Offset: 0

Paul Barry, Feb 12 2003

Values generate solutions to the recurrence a(n) = a(n-1) + k(k+1)* a(n-2), a(0)=1, a(1) = k(k+1)+1. Values and sequences associated with this table are included in A072024.

			Rows are {1}, {1,1,1}, {1,2,2}, {1,3,4,2,1}, {1,4,7,6,3}, ... This is the same as table A035317 with an extra 1 at the end of every second row.
Triangle begins
  1;
  1,  1,  1;
  1,  2,  2;
  1,  3,  4,  2,  1;
  1,  4,  7,  6,  3;
  1,  5, 11, 13,  9,  3,  1;
  1,  6, 16, 24, 22, 12,  4;
  1,  7, 22, 40, 46, 34, 16,  4,  1;
  1,  8, 29, 62, 86, 80, 50, 20,  5;

G. C. Greubel, Rows n = 0..100 of coefficients, flattened

Similar to the triangles A059259, A035317, A108561, A112555. Cf. A059260.

Cf. A001045 (row sums).

Table[CoefficientList[Series[((1+x)^(n+2) -(-1)^n*x^(n+2))/(1+2*x), {x, 0, n+2}], x], {n, 0, 10}]//Flatten (* G. C. Greubel, Feb 18 2019 *)

Rows are generated by P(n,x) = ((x+1)^(n+2) - (-x)^(n+2))/(2*x+1).
The polynomials P(n,-x), n > 0, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re x = 1/2 in the complex plane.
O.g.f.: (1+x*t+x^2*t)/((1+x*t)(1-t-x*t)) = 1 + (1+x+x^2)*t + (1+2x+2x^2)*t^2 + ... . - Peter Bala, Oct 24 2007
T(n,k) = if(k<=2*floor((n+1)/2), Sum_{j=0..floor((n+1)/2)} binomial(n-2j,k-2j), 0). - Paul Barry, Apr 08 2011 (This formula produces the odd numbered rows correctly, but not the even. - G. C. Greubel, Feb 22 2019)

A243201 Odd octagonal numbers indexed by triangular numbers.

Original entry on oeis.org

1, 21, 133, 481, 1281, 2821, 5461, 9633, 15841, 24661, 36741, 52801, 73633, 100101, 133141, 173761, 223041, 282133, 352261, 434721, 530881, 642181, 770133, 916321, 1082401, 1270101, 1481221, 1717633, 1981281, 2274181, 2598421, 2956161, 3349633, 3781141, 4253061, 4767841, 5328001

Offset: 0

Mathew Englander, Jun 01 2014

			a(2) = 133 because the second triangular number is 3 and third odd octagonal number is 133.
a(3) = 481 because the third triangular number is 6 and the sixth odd octagonal number is 481.
a(4) = 1281 because the fourth triangular number is 10 and the tenth odd octagonal number is 1281.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Row 5 of A059259 (coefficients of 1 + 4*n + 7*n^2 + 6*n^3 + 3*n^4 + 0*n^5 which is a formula for the within sequence).
Column 5 of A081297.
Column 6 of A072024.
Diagonal T(n + 1, n) of A219069, n > 0.
Cf. A000217, A000567, A002061, A003215, A005408, A014641, A022522, A140676, A187156.

[3*n^4+6*n^3+7*n^2+4*n+1: n in [0..40]]; // Bruno Berselli, Jun 03 2014

Table[((3 n^2 + 3 n + 2)^2 - 1)/3, {n, 0, 39}] (* Alonso del Arte, Jun 01 2014 *)

[3*n^4+6*n^3+7*n^2+4*n+1 for n in (0..40)] # Bruno Berselli, Jun 03 2014

a(n) = 3*n^4 + 6*n^3 + 7*n^2 + 4*n + 1.
a(n) = (n^2 + n + 1)*(3*n^2 + 3*n + 1).
a(n) = ((3*n^2 + 3*n + 2)^2 - 1)/3.
a(n) = A003215(n) * A002061(n + 1).
a(n) = A022522(n) / A005408(n).
a(n) = A000567(n^2 + n + 1).
a(n) = A014641((n^2 + n)/2).
a(n) = 1 + A140676(n^2 + n).
a(n) = 1 + A187156((n^2 + n + 4)/2) (empirical).
G.f.: (1 + 16*x + 38*x^2 + 16*x^3 + x^4)/(1 - x)^5. - Bruno Berselli, Jun 03 2014
E.g.f.: exp(x)*(1 + 20*x + 46*x^2 + 24*x^3 + 3*x^4). - Stefano Spezia, Apr 16 2022

A072025 a(n) = n^4 + 2n^3 + 4n^2 + 3n + 1 = ((n+1)^5+n^5) / (2n+1).

Original entry on oeis.org

1, 11, 55, 181, 461, 991, 1891, 3305, 5401, 8371, 12431, 17821, 24805, 33671, 44731, 58321, 74801, 94555, 117991, 145541, 177661, 214831, 257555, 306361, 361801, 424451, 494911, 573805, 661781, 759511, 867691, 987041, 1118305, 1262251, 1419671, 1591381

Offset: 0

Henry Bottomley, Jun 06 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A022521, A072024.

[((n+1)^5+n^5)/(2*n+1): n in [0..40]]; // Vincenzo Librandi, Nov 23 2011

Table[((n+1)^5+n^5)/(2n+1),{n,0,30}] (* Vincenzo Librandi, Nov 23 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{1,11,55,181,461},50] (* Harvey P. Dale, Dec 14 2019 *)

a(n)=n^4+2*n^3+4*n^2+3*n+1 \\ Charles R Greathouse IV, Nov 23 2011

Vec((1+x)^2*(1+4*x+x^2)/(1-x)^5 + O(x^100)) \\ Colin Barker, Dec 01 2015

From Colin Barker, Dec 01 2015: (Start)
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>4.
G.f.: (1+x)^2*(1+4*x+x^2) / (1-x)^5.
(End)

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A081297 Array T(k,n), read by antidiagonals: T(k,n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).

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A080242 Table of coefficients of polynomials P(n,x) defined by the relation P(n,x) = (1+x)*P(n-1,x) + (-x)^(n+1).

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A243201 Odd octagonal numbers indexed by triangular numbers.

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A072025 a(n) = n^4 + 2*n^3 + 4*n^2 + 3*n + 1 = ((n+1)^5+n^5) / (2*n+1).

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A072025 a(n) = n^4 + 2n^3 + 4n^2 + 3n + 1 = ((n+1)^5+n^5) / (2n+1).