A074246: Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.

A074246 ↗ nonnegative coefficients

Proved for all n all roots real interlacing (weak)

Recurrence from A074246

\(P_{n}(x) = \left(1 + 2\,x + 2\,n\right)\,P_{n-1}(x) + \left(-x^{2} - 2\,n x - n^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	11	12	3						26
3	50	70	30	4					154
4	274	450	255	60	5				1044
5	1764	3248	2205	700	105	6			8028
6	13068	26264	20307	7840	1610	168	7		69264
7	109584	236248	201852	89796	22680	3276	252	8	663696

All 15 rows b-files: A074246 ?

Technical details

Node id: n-714479c216bddb04 · revision 0 · JSON
Definition source: oeis_catalog:A074246
Fingerprint: f6-792365468f8ca6c1

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A074246 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A001705

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        3,        2;
  2:       11,       12,        3;
  3:       50,       70,       30,        4;
  4:      274,      450,      255,       60,       5;
  5:     1764,     3248,     2205,      700,     105,      6;
  6:    13068,    26264,    20307,     7840,    1610,    168,      7;
  7:   109584,   236248,   201852,    89796,   22680,   3276,    252,     8;
  8:  1026576,  2345400,  2171040,  1077300,  316365,  56700,   6090,   360,   9;
  9: 10628640, 25507152, 25228500, 13667720, 4510275, 946638, 127050, 10560, 495, 10;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+1+2*x)*P(n-1,x) - (n^2+2*n*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2*n + 1 + 2*x)*P[n - 1, x] - (n^2 + 2*n*x + x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2*n + 1 + 2*x)*v[i - 1] - (n^2 + 2*n*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A074246 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:        1;
   2:        3,        2;
   3:       11,       12,        3;
   4:       50,       70,       30,        4;
   5:      274,      450,      255,       60,       5;
   6:     1764,     3248,     2205,      700,     105,      6;
   7:    13068,    26264,    20307,     7840,    1610,    168,      7;
   8:   109584,   236248,   201852,    89796,   22680,   3276,    252,     8;
   9:  1026576,  2345400,  2171040,  1077300,  316365,  56700,   6090,   360,   9;
  10: 10628640, 25507152, 25228500, 13667720, 4510275, 946638, 127050, 10560, 495, 10;

%F Formula

The row polynomials satisfy P(n,x) = (2*n-1+2*x)*P(n-1,x) - (n^2-2*n+1+(2*n-2)*x+x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + 2*x)*P[n - 1, x] - (n^2 - 2*n + 1 + (2*n - 2)*x + x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (2*n - 1 + 2*x)*v[i - 1] - (n^2 - 2*n + 1 + (2*n - 2)*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, +∞)
by Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A130534 → D

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003