A174531 -id:A174531 - cp's OEIS frontend

A174625 Table T(n,k) with the coefficients of the polynomial P_n(x) = P_{n-1}(x) + x*P_{n-2}(x) + 1 in row n, by decreasing exponent of x.

0, 2, 3, 2, 4, 5, 5, 2, 9, 6, 7, 14, 7, 2, 16, 20, 8, 9, 30, 27, 9, 2, 25, 50, 35, 10, 11, 55, 77, 44, 11, 2, 36, 105, 112, 54, 12, 13, 91, 182, 156, 65, 13, 2, 49, 196, 294, 210, 77, 14, 15, 140, 378, 450, 275, 90, 15, 2, 64, 336, 672, 660, 352, 104, 16, 17, 204, 714, 1122, 935, 442

Offset: 1

Vladimir Shevelev, Mar 24 2010

The polynomials are defined by the recurrence starting with P_1(x)=0, P_2(x)=2.
The degree of the polynomial (row length minus 1) is A004526(n-2).
All coefficients of P_n are multiples of n iff n is prime.
Apparently a mirrored version of A157000. [R. J. Mathar, Nov 01 2010]

			The table starts
0; # 0
2; # 2
3; # 3
2,4; # 4+2*x
5,5; # 5+5*x
2,9,6; # 6+9*x+2*x^2
7,14,7; # 7+14*x+7*x^2
2,16,20,8; # 8+20*x+16*x^2+2*x^3
9,30,27,9; # 9+27*x+30*x^2+9*x^3
2,25,50,35,10; # 10+35*x+50*x^2+25*x^3+2*x^4
11,55,77,44,11; # 11+44*x+77*x^2+55*x^3+11*x^4

Cf. A018187, A013998, A174531.

p[0]:=0 p[1]:=2; p[n_]:=p[n]=Expand[p[n-1] +x p[n-2]+1]; Flatten[{0, Map[Reverse[CoefficientList[#,x]]&, Table[Expand[p[n]], {n,0,20}]]}] (* Peter J. C. Moses, Aug 18 2013 *)

Definition rephrased, sequence extended, keyword:tabf, examples added R. J. Mathar, Nov 01 2010

A185139 Triangle T(n,k) = Sum_{i=1..n} 2^(i-1)C(n+2k-i-1, k-1), 1 <= k <= n.

Original entry on oeis.org

1, 3, 10, 7, 25, 91, 15, 56, 210, 792, 31, 119, 456, 1749, 6721, 63, 246, 957, 3718, 14443, 56134, 127, 501, 1969, 7722, 30251, 118456, 463828, 255, 1012, 4004, 15808, 62322, 245480, 966416, 3803648, 511, 2035, 8086, 32071, 127024, 502588, 1987096, 7852453, 31020445, 1023, 4082, 16263

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Feb 04 2012

The first term of the m-th row is 2^m-1.

			Triangle begins
1,
3,     10,
7,     25,    91,
15,    56,    210,  792,
31,    119,   456,  1749,  6721,
63,    246,   957,  3718,  14443,  56134,
127,   501,   1969, 7722,  30251,  118456, 463828,
255,   1012,  4004, 15808, 62322,  245480, 966416,  3803648,
511,   2035,  8086, 32071, 127024, 502588, 1987096, 7852453, 31020445,
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
V. Shevelev and P. Moses, On a sequence of polynomials with hypothetically integer coefficients arXiv:1112.5715 [math.NT], 2011.

Cf. A174531.

Table[Sum[2^(j - 1)*Binomial[n + 2*k - j - 1, k - 1], {j, 1, n}], {n,
   1, 50}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 23 2017 *)

for(n=1,20, for(k=1,n, print1(sum(j=1,n, 2^(j-1)*binomial(n+2*k-j-1,k-1)), ", "))) \\ G. C. Greubel, Jun 23 2017

2*T_n(k) = T_(n-1)(k+1) + C(n+2*k-1,k).
T_n(k) = T_(n-2)(k+1) + C(n+2*k-1,k).
T_n(k) = 2*T_(n-1)(k) + C(n+2*k-2,k-1).
T_n(k+1) = 4*T_n(k) - (n/k)*C(n+2*k-1,k-1).

A195609 Numbers n such that Sum_{i=1..n} A(i) = A(n)*A(n+1)/4, where A(n) = A000069(n).

Original entry on oeis.org

3, 4, 5, 9, 15, 16, 17, 23, 27, 28, 29, 33, 39, 43, 44, 45, 51, 52, 53, 57, 63, 64, 65, 71, 75, 76, 77, 83, 84, 85, 89, 95, 99, 100, 101, 105, 111, 112, 113, 119, 123, 124, 125, 129, 135, 139, 140, 141, 147, 148, 149, 153, 159, 163, 164, 165, 169, 175, 176

Offset: 1

Vladimir Shevelev, Sep 21 2011

nonn

Conjectures: 1) there are only 3 different first differences 1,4,6; 2) the sequence contains either isolated series of terms, e.g., {9},{23},{33},{39},..., or series of 3 consecutive integers, e.g., {3,4,5}, {15,16,17}, etc.; 3)the first terms m of every series satisfy the condition A(m+1)-A(m-1)=5, where A(n)=A000069(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000069, A174531, A001969, A195608.

a = Select[Range[1000], OddQ[DigitCount[#, 2][[1]]] &]; t = {}; s = 0; Do[s = s + a[[n]]; If[s == a[[n]] a[[n + 1]]/4, AppendTo[t, n]], {n, Length[a] - 1}]; t (* T. D. Noe, Sep 23 2011 *)

cp's OEIS Frontend

A174625 Table T(n,k) with the coefficients of the polynomial P_n(x) = P_{n-1}(x) + x*P_{n-2}(x) + 1 in row n, by decreasing exponent of x.

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A185139 Triangle T(n,k) = Sum_{i=1..n} 2^(i-1)C(n+2k-i-1, k-1), 1 <= k <= n.

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A195609 Numbers n such that Sum_{i=1..n} A(i) = A(n)*A(n+1)/4, where A(n) = A000069(n).

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cp's OEIS Frontend

A174625 Table T(n,k) with the coefficients of the polynomial P_n(x) = P_{n-1}(x) + x*P_{n-2}(x) + 1 in row n, by decreasing exponent of x.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A185139 Triangle T(n,k) = Sum_{i=1..n} 2^(i-1)*C(n+2*k-i-1, k-1), 1 <= k <= n.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

PARI

Formula

A195609 Numbers n such that Sum_{i=1..n} A(i) = A(n)*A(n+1)/4, where A(n) = A000069(n).

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

A185139 Triangle T(n,k) = Sum_{i=1..n} 2^(i-1)C(n+2k-i-1, k-1), 1 <= k <= n.