A257615 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 12\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-225 - 348\,x + 369\,x^{2} - 240\,n - 1016\,n x + 1392\,n x^{2} - 60\,n^{2} - 720\,n^{2} x + 1688\,n^{2} x^{2} - 160\,n^{3} x + 832\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-384\,x + 1284\,x^{2} - 900\,x^{3} - 320\,n x + 1832\,n x^{2} - 1272\,n x^{3} - 80\,n^{2} x + 944\,n^{2} x^{2} - 352\,n^{2} x^{3} + 160\,n^{3} x^{2} + 160\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-84\,x^{2} + 408\,x^{3} - 324\,x^{4} - 64\,n x^{2} + 640\,n x^{3} - 576\,n x^{4} - 16\,n^{2} x^{2} + 352\,n^{2} x^{3} - 336\,n^{2} x^{4} + 64\,n^{3} x^{3} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(90 + 108\,x + 108\,n + 216\,n x + 32\,n^{2} + 144\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(60\,x - 180\,x^{2} + 56\,n x - 312\,n x^{2} + 16\,n^{2} x - 176\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 5 + 20\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	20														25
2	25	280	225													530
3	125	2860	6435	2000												11420
4	625	25520	115830	102080	15625											259680
5	3125	211140	1674090	2976160	1319625	112500										6296640
6	15625	1665480	21242655	65475520	59007375	14989320	765625									163161600
7	78125	12720860	247393845	1211172080	1892456375	989575380	155830535	5000000								4514227200
8	390625	94983520	2713117500	19920062080	49203865750	44820139680	14771417500	1519736320	31640625							133075353600
9	1953125	697697140	28478602980	301117122880	1106526709750	1593398462040	922171188820	202514510080	14128367085	195312500						4169229926400
10	9765625	5063567480	289273616685	4274232274560	22383270803250	47801953861200	43871210774370	17096929098240	2603462550165	126589187000	1181640625					138453177139200
11	48828125	36421534860	2865502167255	57816103376400	417873791093250	1264971777181560	1721767141163790	1069756905198720	292694023343025	31838912969500	1101751429515	7031250000				4860729409536000
12	244140625	260224181520	27838230841170	753144336868160	7329450176250375	30407603582034720	58540082799568860	54057961923617280	23747418571051215	4707152105426000	374269547975730	9368070534720	41259765625			179954591072256000
13	1220703125	1849889583140	266352696597870	9521842409601760	122348976897454375	677756627060332140	1781555822450290740	2326930053812624640	1524952410885747315	489395907589817500	72008933222613310	4261643145565920	78157834887665	239257812500		7009078818373632000
14	6103515625	13100594269480	2517879564680715	117525195942317120	1962483668791975625	14218547520748671000	49658992571796102915	88423342044879736320	82089355067662945635	39495965335412975000	9498420956953162025	1057726763480854080	47280182936782315	641929119204520	1373291015625	286572813597278208000

Technical details

Node id: n-41fe3e2d4341ffdb · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-ea23bfbe9c2dc6f1

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:       1;
  1:       5,        20;
  2:      25,       280,         225;
  3:     125,      2860,        6435,         2000;
  4:     625,     25520,      115830,       102080,         15625;
  5:    3125,    211140,     1674090,      2976160,       1319625,        112500;
  6:   15625,   1665480,    21242655,     65475520,      59007375,      14989320,       765625;
  7:   78125,  12720860,   247393845,   1211172080,    1892456375,     989575380,    155830535,      5000000;
  8:  390625,  94983520,  2713117500,  19920062080,   49203865750,   44820139680,  14771417500,   1519736320,    31640625;
  9: 1953125, 697697140, 28478602980, 301117122880, 1106526709750, 1593398462040, 922171188820, 202514510080, 14128367085, 195312500;

%F Formula

The row polynomials satisfy (4*n^2+12*n+9)*P(n,x) = -(60*n^2+240*n+225+(160*n^3+720*n^2+1016*n+348)*x-(144*n^4+832*n^3+1688*n^2+1392*n+369)*x^2)*P(n-2,x) - ((80*n^2+320*n+384)*x-(160*n^3+944*n^2+1832*n+1284)*x^2-(64*n^4+160*n^3-352*n^2-1272*n-900)*x^3)*d/dx P(n-2,x) - ((16*n^2+64*n+84)*x^2-(64*n^3+352*n^2+640*n+408)*x^3+(64*n^3+336*n^2+576*n+324)*x^4)*d^2/dx^2 P(n-2,x) + (32*n^2+108*n+90+(32*n^3+144*n^2+216*n+108)*x)*P(n-1,x) + ((16*n^2+56*n+60)*x-(32*n^3+176*n^2+312*n+180)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 20*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 20*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(60*n^2 + 240*n + 225 + (160*n^3 + 720*n^2 + 1016*n + 348)*x - (144*n^4 + 832*n^3 + 1688*n^2 + 1392*n + 369)*x^2)*P[n - 2, x] - ((80*n^2 + 320*n + 384)*x - (160*n^3 + 944*n^2 + 1832*n + 1284)*x^2 - (64*n^4 + 160*n^3 - 352*n^2 - 1272*n - 900)*x^3)*D[P[n - 2, x], x] - ((16*n^2 + 64*n + 84)*x^2 - (64*n^3 + 352*n^2 + 640*n + 408)*x^3 + (64*n^3 + 336*n^2 + 576*n + 324)*x^4)*D[P[n - 2, x], {x, 2}] + (32*n^2 + 108*n + 90 + (32*n^3 + 144*n^2 + 216*n + 108)*x)*P[n - 1, x] + ((16*n^2 + 56*n + 60)*x - (32*n^3 + 176*n^2 + 312*n + 180)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 12*n + 9)]];
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] = 5 + 20*x; for(n = 2, N, my(i = n + 1); v[i] = (-(60*n^2 + 240*n + 225 + (160*n^3 + 720*n^2 + 1016*n + 348)*x - (144*n^4 + 832*n^3 + 1688*n^2 + 1392*n + 369)*x^2)*v[i - 2] - ((80*n^2 + 320*n + 384)*x - (160*n^3 + 944*n^2 + 1832*n + 1284)*x^2 - (64*n^4 + 160*n^3 - 352*n^2 - 1272*n - 900)*x^3)*deriv(v[i - 2], x) - ((16*n^2 + 64*n + 84)*x^2 - (64*n^3 + 352*n^2 + 640*n + 408)*x^3 + (64*n^3 + 336*n^2 + 576*n + 324)*x^4)*deriv(deriv(v[i - 2], x), x) + (32*n^2 + 108*n + 90 + (32*n^3 + 144*n^2 + 216*n + 108)*x)*v[i - 1] + ((16*n^2 + 56*n + 60)*x - (32*n^3 + 176*n^2 + 312*n + 180)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 12*n + 9)); 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 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.?
    
    ↑ Liu–Wang theorem (above)

References

[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.