A099040 → (θ+1)²

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(2\,n + 3\,n^{2} - 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(4 + 16\,x - 8\,x^{2} + 2\,n + 22\,n x - 8\,n x^{2} - 4\,n^{2} + 10\,n^{2} x - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x - 4\,x^{2} - 4\,x^{3} + 12\,n x - 2\,n x^{2} - 4\,n x^{3} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	8														8
2	0	8	36													44
3	0	0	72	128												200
4	0	0	36	384	400											820
5	0	0	0	384	1600	1152										3136
6	0	0	0	128	2400	5760	3136									11424
7	0	0	0	0	1600	11520	18816	8192								40128
8	0	0	0	0	400	11520	47040	57344	20736							137040
9	0	0	0	0	0	5760	62720	172032	165888	51200						457600
10	0	0	0	0	0	1152	47040	286720	580608	460800	123904					1500224
11	0	0	0	0	0	0	18816	286720	1161216	1843200	1239040	294912				4843904
12	0	0	0	0	0	0	3136	172032	1451520	4300800	5575680	3244032	692224			15439424
13	0	0	0	0	0	0	0	57344	1161216	6451200	14868480	16220160	8306688	1605632		48670720
14	0	0	0	0	0	0	0	8192	580608	6451200	26019840	48660480	45686784	20873216	3686400	151966720

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Technical details

Node id: n-eee84e53bbac8147 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d1753e4390bc8e4c

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 0, 8;
  2: 0, 8, 36;
  3: 0, 0, 72, 128;
  4: 0, 0, 36, 384,  400;
  5: 0, 0,  0, 384, 1600,  1152;
  6: 0, 0,  0, 128, 2400,  5760,  3136;
  7: 0, 0,  0,   0, 1600, 11520, 18816,   8192;
  8: 0, 0,  0,   0,  400, 11520, 47040,  57344,  20736;
  9: 0, 0,  0,   0,    0,  5760, 62720, 172032, 165888, 51200;

%F Formula

The row polynomials satisfy (n^3+3*n^2+2*n-2*n^2*x)*P(n,x) = -(2*n^3+4*n^2-2*n-4-(10*n^2+22*n+16)*x+(8*n+8)*x^2)*P(n-1,x) + ((4*n^2+12*n+8)*x+(2*n^2-2*n-4)*x^2-(4*n+4)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 4*n^2 - 2*n - 4 - (10*n^2 + 22*n + 16)*x + (8*n + 8)*x^2)*P[n - 1, x] + ((4*n^2 + 12*n + 8)*x + (2*n^2 - 2*n - 4)*x^2 - (4*n + 4)*x^3)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n - 2*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; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 + 4*n^2 - 2*n - 4 - (10*n^2 + 22*n + 16)*x + (8*n + 8)*x^2)*v[i - 1] + ((4*n^2 + 12*n + 8)*x + (2*n^2 - 2*n - 4)*x^2 - (4*n + 4)*x^3)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n - 2*n^2*x)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-974408a4688a73a4
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-974408a4688a73a4
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-974408a4688a73a4

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image euler_polynomial, shift 0 e-974408a4688a73a4

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Riordan array (1, 2+2x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - basis_map A099040 e-52040eb0dce100d3
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
  - theorem_check ma_wang_interval e-17786677d2ed021b root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
  - basis_map A027467 e-0a457ba83443136e
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
    
    preserver_image affine, shift 0 e-48e4dfa7d694f880
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038255 e-162624a94908f408
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
    
    preserver_image affine, shift 0 e-79e1a4a5e521405c
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A027466 e-3f03a2606d2e6662
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-5e00209341210844
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038303 e-532ccda39f04a0f0
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
    
    preserver_image affine, shift 0 e-c2f106499058af8f
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038207 e-72107609b2d5e6bc
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
    
    preserver_image affine, shift 0 e-8c4e4f825f4ab306
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038243 e-763de8a298cf812d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
    
    preserver_image affine, shift 0 e-bdc3e5aca0457f81
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038279 e-8232f658d776d13c
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
    
    preserver_image affine, shift 0 e-4478346993b2c814
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A027465 e-af79ef336fd9f8da
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
    
    preserver_image affine, shift 0 e-319b2a76ea6c1ab0
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038291 e-c3d762863d6658d7
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
    
    preserver_image affine, shift 0 e-b9371054cfe1d371
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A007318 e-e158703ce212fb71
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
    
    preserver_image affine, shift 0 e-dc7a132d665751c2
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038231 e-e36b44dfb8fba667
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
    
    preserver_image affine, shift 0 e-23d8b9b067cbe12f
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038327 e-ffeb0648278e3cee
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
    
    preserver_image affine, shift 0 e-1000187f132dd1ea
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.