A038329 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 100\,x + 64\,x^{2} + 60\,n x + 64\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(375 + 2700\,x + 4080\,x^{2} + 1728\,x^{3} + 900\,n x + 2160\,n x^{2} + 1152\,n x^{3} + 240\,n^{2} x^{2} + 192\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 225 + 1440\,x + 1296\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	225	1440	1296															2961
1	3375	32400	58320	27648														121743
2	50625	648000	1749600	1658880	518400													4625505
3	759375	12150000	43740000	62208000	38880000	8957952												166695327
4	11390625	218700000	984150000	1866240000	1749600000	806215680	146313216											5782609521
5	170859375	3827250000	20667150000	48988800000	61236000000	42326323200	15362887680	2293235712										194872505967
6	2562890625	65610000000	413343000000	1175731200000	1837080000000	1693052928000	921773260800	275188285440	34828517376									6419170082241
7	38443359375	1107168750000	7971615000000	26453952000000	49601160000000	57140536320000	41479796736000	18575209267200	4701849845760	515978035200								207585709313535
8	576650390625	18452812500000	149467781250000	566870400000000	1240029000000000	1714216089600000	1555492377600000	928760463360000	352638738432000	77396705280000	7492001071104							6611393019483729
9	8649755859375	304471406250000	2740242656250000	11691702000000000	29229255000000000	47140942464000000	51331248460800000	38311369113600000	19395130613760000	6385228185600000	1236180176732160	106993205379072						207881413038230607
10	129746337890625	4982259375000000	49324367812500000	233834040000000000	657658237500000000	1212195663360000000	1539937453824000000	1379209288089600000	872780877619200000	383113691136000000	111256215905894400	19258776968232960	1506820975755264					6465187438904073249
11	1946195068359375	80961714843750000	874386520312500000	4559763780000000000	14249261812500000000	29547269294400000000	42898257642240000000	44824301862912000000	34038454227148800000	18676792442880000000	7231654033883136000	1877730754402713600	293830090272276480	20970668254298112				199175581039117833567
12	29192926025390625	1307843085937500000	15301764105468750000	87050035800000000000	299234498062500000000	689436283536000000000	1126079263108800000000	1344729055887360000000	1191345897950208000000	784425282600960000000	379661836778864640000	131441152808189952000	30852159478589030400	4403840333402603520	288881654523494400			6085586988116829360945
13	437893890380859375	21018906738281250000	264838224902343750000	1632188171250000000000	6120705642187500000000	15512316379560000000000	28151981577720000000000	37820504696832000000000	38293261005542400000000	29415948097536000000000	17084782655048908800000	7393564845460684800000	2313911960894177280000	495432037507792896000	64998372267786240000	3944197523094110208		184589834664861349985583
14	6568408355712890625	336302507812500000000	4540083855468750000000	30132704700000000000000	122414112843750000000000	338450539190400000000000	675647557865280000000000	1008546791915520000000000	1148797830166272000000000	1008546791915520000000000	683391306201956352000000	354891112582112870400000	138834717653650636800000	39634563000623431680000	7799804672134348800000	946607405542586449920	53431550820665524224	5562970826435219854544769

Technical details

Node id: n-e782416b20ea362c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7f34919e5b06c424

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:           225,            1440,             1296;
  1:          3375,           32400,            58320,             27648;
  2:         50625,          648000,          1749600,           1658880,            518400;
  3:        759375,        12150000,         43740000,          62208000,          38880000,           8957952;
  4:      11390625,       218700000,        984150000,        1866240000,        1749600000,         806215680,         146313216;
  5:     170859375,      3827250000,      20667150000,       48988800000,       61236000000,       42326323200,       15362887680,        2293235712;
  6:    2562890625,     65610000000,     413343000000,     1175731200000,     1837080000000,     1693052928000,      921773260800,      275188285440,       34828517376;
  7:   38443359375,   1107168750000,    7971615000000,    26453952000000,    49601160000000,    57140536320000,    41479796736000,    18575209267200,     4701849845760,     515978035200;
  8:  576650390625,  18452812500000,  149467781250000,   566870400000000,  1240029000000000,  1714216089600000,  1555492377600000,   928760463360000,   352638738432000,   77396705280000,    7492001071104;
  9: 8649755859375, 304471406250000, 2740242656250000, 11691702000000000, 29229255000000000, 47140942464000000, 51331248460800000, 38311369113600000, 19395130613760000, 6385228185600000, 1236180176732160, 106993205379072;

%F Formula

The row polynomials satisfy (25+(60*n+100)*x+(16*n^2+64*n+64)*x^2)*P(n,x) = (375+(900*n+2700)*x+(240*n^2+2160*n+4080)*x^2+(192*n^2+1152*n+1728)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 1440*x + 1296*x^2.

%t Mathematica

P[0, x_] := 225 + 1440*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((375 + (900*n + 2700)*x + (240*n^2 + 2160*n + 4080)*x^2 + (192*n^2 + 1152*n + 1728)*x^3)*P[n - 1, x])/(25 + (60*n + 100)*x + (16*n^2 + 64*n + 64)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 225 + 1440*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((375 + (900*n + 2700)*x + (240*n^2 + 2160*n + 4080)*x^2 + (192*n^2 + 1152*n + 1728)*x^3)*v[i - 1])/(25 + (60*n + 100)*x + (16*n^2 + 64*n + 64)*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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

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.