A038331 → basis A013609 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(289 + 850\,x + 400\,x^{2} + 510\,n x + 400\,n x^{2} + 100\,n^{2} x^{2}\right)\,P_{n}(x) = \left(4913 + 26010\,x + 28900\,x^{2} + 9000\,x^{3} + 8670\,n x + 15300\,n x^{2} + 6000\,n x^{3} + 1700\,n^{2} x^{2} + 1000\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 289 + 1360\,x + 900\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	289	1360	900															2549
1	4913	34680	45900	16000														101493
2	83521	786080	1560600	1088000	250000													3768201
3	1419857	16704200	44217000	46240000	21250000	3600000												133431057
4	24137569	340765680	1127533500	1572160000	1083750000	367200000	49000000											4564546749
5	410338673	6758519320	26835297300	46771760000	42988750000	21848400000	5831000000	640000000										152084065293
6	6975757441	131308375360	608266738800	1272191872000	1461617500000	990460800000	396508000000	87040000000	8100000000									4962469043601
7	118587876497	2511272678760	13294973005200	32440892736000	44725495500000	37885125600000	20221908000000	6658560000000	1239300000000	100000000000								159196115396457
8	2015993900449	47435150598800	282518176360500	787850252160000	1267222372500000	1288094270400000	859431090000000	377318400000000	105340500000000	17000000000000	1210000000000							5035436205919749
9	34271896307633	887037316197560	5870099886601500	18415999644240000	33852940522500000	40145604760800000	32142722766000000	17639635200000000	6566224500000000	1589500000000000	226270000000000	14400000000000						157384706492646693
10	582622237229761	16450510227663840	119750037686670600	417429325269440000	863249983323750000	1169957624457600000	1092852574044000000	719697116160000000	334877449500000000	108086000000000000	23079540000000000	2937600000000000	169000000000000					4869119382906354201
11	9904578032905937	302963563359475720	2405887120795836600	9225188088454624000	21197582923838750000	32320079375641200000	34502916980532000000	26508843778560000000	14801583267900000000	5971751500000000000	1700192780000000000	324604800000000000	37349000000000000	1960000000000000				149310807757114792257
12	168377826559400929	5546563698427324720	47716761229117425900	199599524095654592000	504502473587362250000	854686543489178400000	1026461780170827000000	901300688471040000000	587129469626700000000	284255371400000000000	101161470410000000000	25751980800000000000	4444531000000000000	466480000000000000	22500000000000000			4543214515804866393549
13	2862423051509815793	101026695935640557400	935982624109611046500	4241489887032660080000	11695284614979761250000	21794506858974049200000	29083083771506765000000	28728959445014400000000	21388287822115500000000	12080853284500000000000	5159234990910000000000	1641688776000000000000	377785135000000000000	59476200000000000000	5737500000000000000	256000000000000000		137296516029129896949693
14	48661191875666868481	1831950752966282107520	18184805268415300332000	88745019174837195520000	265093117939541255000000	538918715058267398400000	791059878584984008000000	868252996560435200000000	727201785951927000000000	469427441912000000000000	233885319587920000000000	89307869414400000000000	25689389180000000000000	5392508800000000000000	780300000000000000000	69632000000000000000	2890000000000000000	4123892281377569306228001

Technical details

Node id: n-daa6779e2a7ff82e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-36574f7e4cd1fbfa

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:            289,            1360,              900;
  1:           4913,           34680,            45900,             16000;
  2:          83521,          786080,          1560600,           1088000,            250000;
  3:        1419857,        16704200,         44217000,          46240000,          21250000,           3600000;
  4:       24137569,       340765680,       1127533500,        1572160000,        1083750000,         367200000,          49000000;
  5:      410338673,      6758519320,      26835297300,       46771760000,       42988750000,       21848400000,        5831000000,         640000000;
  6:     6975757441,    131308375360,     608266738800,     1272191872000,     1461617500000,      990460800000,      396508000000,       87040000000,       8100000000;
  7:   118587876497,   2511272678760,   13294973005200,    32440892736000,    44725495500000,    37885125600000,    20221908000000,     6658560000000,    1239300000000,     100000000000;
  8:  2015993900449,  47435150598800,  282518176360500,   787850252160000,  1267222372500000,  1288094270400000,   859431090000000,   377318400000000,  105340500000000,   17000000000000,   1210000000000;
  9: 34271896307633, 887037316197560, 5870099886601500, 18415999644240000, 33852940522500000, 40145604760800000, 32142722766000000, 17639635200000000, 6566224500000000, 1589500000000000, 226270000000000, 14400000000000;

%F Formula

The row polynomials satisfy (289+(510*n+850)*x+(100*n^2+400*n+400)*x^2)*P(n,x) = (4913+(8670*n+26010)*x+(1700*n^2+15300*n+28900)*x^2+(1000*n^2+6000*n+9000)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1360*x + 900*x^2.

%t Mathematica

P[0, x_] := 289 + 1360*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4913 + (8670*n + 26010)*x + (1700*n^2 + 15300*n + 28900)*x^2 + (1000*n^2 + 6000*n + 9000)*x^3)*P[n - 1, x])/(289 + (510*n + 850)*x + (100*n^2 + 400*n + 400)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1360*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4913 + (8670*n + 26010)*x + (1700*n^2 + 15300*n + 28900)*x^2 + (1000*n^2 + 6000*n + 9000)*x^3)*v[i - 1])/(289 + (510*n + 850)*x + (100*n^2 + 400*n + 400)*x^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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.