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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	1026576	2345400	2171040	1077300	316365	56700	6090	360	9							6999840
9	10628640	25507152	25228500	13667720	4510275	946638	127050	10560	495	10						80627040
10	120543840	301835952	315774228	183982920	66697675	15825348	2501961	261360	17325	660	11					1007441280
11	1486442880	3863119104	4242044664	2628827344	1030350750	269941386	48486438	5995704	501930	27170	858	12				13575738240
12	19802759040	53193434112	60941259288	39830815024	16680593930	4745658918	945255311	133357224	13270257	910910	41041	1092	13			196287356160
13	283465647360	784313595648	932967781200	638886422720	283316833800	86455937568	18770176425	2947292920	335810475	27497470	1576575	60060	1365	14		3031488633600
14	4339163001600	12331635229440	15170987111472	10827253382400	5048360535400	1636819264080	382417906871	65661024000	8352861660	785584800	53895842	2620800	85540	1680	15	49811492505600

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