A025169 -id:A025169 - cp's OEIS frontend

A185331 Riordan array ((1-x+x^2)/(1+x^2), x/(1+x^2)).

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 0, 2, -2, -1, 1, -1, 1, 3, -3, -1, 1, 0, -3, 3, 4, -4, -1, 1, 1, -1, -6, 6, 5, -5, -1, 1, 0, 4, -4, -10, 10, 6, -6, -1, 1, -1, 1, 10, -10, -15, 15, 7, -7, -1, 1, 0, -5, 5, 20, -20, -21, 21, 8, -8, -1, 1

Offset: 0

Philippe Deléham, Feb 08 2012

Triangle T(n,k), read by rows, given by (-1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins:
   1;
  -1,  1;
   0, -1,   1;
   1, -1,  -1,   1;
   0,  2,  -2,  -1,   1;
  -1,  1,   3,  -3,  -1,   1;
   0, -3,   3,   4,  -4,  -1,   1;
   1, -1,  -6,   6,   5,  -5,  -1,  1;
   0,  4,  -4, -10,  10,   6,  -6, -1,  1;
  -1,  1,  10, -10, -15,  15,   7, -7, -1,  1;
   0, -5,   5,  20, -20, -21,  21,  8, -8, -1,  1;
   1, -1, -15,  15,  35, -35, -28, 28,  9, -9, -1, 1;

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

Cf. A206474 (unsigned version).

Cf. A007318, A053122, A078812, A085478, A128908, A129818,

CoefficientList[Series[CoefficientList[Series[(1 - x + x^2)/(1 - y*x + x^2), {x, 0, 10}], x], {y, 0, 10}], y] // Flatten (* G. C. Greubel, Jun 27 2017 *)

T(n,k) = T(n-1,k-1) - T(n-2,k), T(0,0) = 1, T(0,1) = -1, T(0,2) = 0.
G.f.: (1-x+x^2)/(1-y*x+x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A184334(n), A163805(n), A000007(n), A028310(n), A025169(n-1), A005320(n) (n>0) for x = -1, 0, 1, 2, 3, 4 respectively.
T(n,n) = 1, T(n+1,n) = -1, T(n+2,n) = -n, T(n+3,n) = n+1, T(n+4,n) = n(n+1)/2 = A000217(n).
T(2n,2k) = (-1)^(n-k) * A128908(n,k), T(2n+1,2k+1) = -T(2n+1,2k) = A129818(n,k), T(2n+2,2k+1) = (-1)*A053122(n,k). - Philippe Deléham, Feb 09 2012

A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 4, 14, 16, 7, 1, 5, 25, 41, 29, 9, 1, 6, 41, 91, 92, 46, 11, 1, 7, 63, 182, 246, 175, 67, 13, 1, 8, 92, 336, 582, 550, 298, 92, 15, 1, 9, 129, 582, 1254, 1507, 1079, 469, 121, 17, 1, 10, 175, 957, 2508, 3718, 3367, 1925, 696, 154

Offset: 0

Philippe Deléham, Jan 24 2014

Triangle T(n,k), read by rows, given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A111282(n+1) = A025169(n-1).
Diagonal sums are A122391(n+1) = A003945(n-1).

			Triangle begins:
  1;
  1,  1;
  2,  3,   1;
  3,  7,   5,   1;
  4, 14,  16,   7,   1;
  5, 25,  41,  29,   9,  1;
  6, 41,  91,  92,  46, 11,  1;
  7, 63, 182, 246, 175, 67, 13, 1;

Cf. Columns: A028310, A004006.
Cf. Diagonals: A000012, A005408, A130883.
Cf. Similar sequences: A078812, A085478, A111125, A128908, A165253, A207606.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare(1+x/(1-x)^2, 8); # Peter Luschny, Mar 06 2022

CoefficientList[#, y] & /@
CoefficientList[
Series[(1 - x + x^2)/(1 - 2*x - x*y + x^2), {x, 0, 12}], x] (* Wouter Meeussen, Jan 25 2014 *)

G.f.: (1 - x + x^2)/(1 - 2*x - x*y + x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or k > n.
The Riordan square (see A321620) of 1 + x/(1 - x)^2. - Peter Luschny, Mar 06 2022

A358464 a(n) is the greatest m such that Sum_{k = 1..m} 1/(1 + n*k) <= 1.

Original entry on oeis.org

2, 6, 16, 42, 110, 288, 761, 2020, 5388, 14417, 38681, 103994, 280032, 755031, 2037848, 5504884, 14880978, 40250609, 108926101, 294902398, 798703663, 2163878141

Offset: 1

Thomas Scheuerle, Nov 18 2022

This sequence coincidences with 2*Fibonacci(2*n) (A025169) for the first 6 terms.

			a(2) = 6 because Sum_{m = 1..a(2)} 1/(1+2*m) = 43024/45045 < 1, but Sum_{m = 1..a(2)+1} 1/(1+2*m) = 46027/45045 > 1.

David Borwein and Jonathan M. Borwein, Some Remarkable Properties of Sinc and Related Integrals, The Ramanujan Journal, 5 (2001) 73 - 89.
Hanspeter Schmid, Two curious integrals and a graphic proof, Elemente der Mathematik, 69 (2014) 11 - 17.

Cf. A002387, A025169, A064169.

a(n) = {my(k = 2*fibonacci(2*n)-1);my(b = (psi(k+(1/n))-psi(1+(1/n)))/n); until(b > 1, b = b+(1/(1+n*k)); k=k+1 );k-2}

ceiling(digamma(a(n)+(1/n)+1) - digamma((1/n)+1)) = n.
Integral_{x=0..oo} Product_{k=0..m} sinc(x/(n*k+1)) dx = Pi for 0 <= m <= a(n). See links Schmid and Borwein.
ceiling(Sum_{m = 0..oo} ( 1/(m+1) * Sum_{k = 0..m} (-1)^k*binomial(m, k)*log( (a(n)+(1/n)+1+k) / ((1/n)+1+k) ) )) = n.
a(n) ~ floor(exp(n + digamma(1+(1/n))) - (1/2) - (1/n)). This appears to be accurate for at least n < 22.

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A185331 Riordan array ((1-x+x^2)/(1+x^2), x/(1+x^2)).

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A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

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A358464 a(n) is the greatest m such that Sum_{k = 1..m} 1/(1 + n*k) <= 1.

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