A113214 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2} - n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-2\,x - 2\,x^{2} + 4\,n + 3\,n x - 2\,n x^{2} + 2\,n^{2} + 5\,n^{2} x - 2\,n^{3}\right)\,P_{n-1}(x) + \left(-4\,x^{2} - x^{3} + 4\,n x - 3\,n x^{2} - n x^{3} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	0	12	9													21
3	0	8	36	16												60
4	0	0	45	80	25											150
5	0	0	18	144	150	36										348
6	0	0	0	112	350	252	49									763
7	0	0	0	32	400	720	392	64								1608
8	0	0	0	0	225	1080	1323	576	81							3285
9	0	0	0	0	50	900	2450	2240	810	100						6550
10	0	0	0	0	0	396	2695	4928	3564	1100	121					12804
11	0	0	0	0	0	72	1764	6720	9072	5400	1452	144				24624
12	0	0	0	0	0	0	637	5824	14742	15600	7865	1872	169			46709
13	0	0	0	0	0	0	98	3136	15876	29400	25410	11088	2366	196		87570
14	0	0	0	0	0	0	0	960	11340	37800	54450	39600	15210	2940	225	162525

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Technical details

Node id: n-1f4d09521acf5bb3 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-8c59810c7b2f08c6

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: 2,  4;
  2: 0, 12,  9;
  3: 0,  8, 36,  16;
  4: 0,  0, 45,  80,  25;
  5: 0,  0, 18, 144, 150,   36;
  6: 0,  0,  0, 112, 350,  252,   49;
  7: 0,  0,  0,  32, 400,  720,  392,   64;
  8: 0,  0,  0,   0, 225, 1080, 1323,  576,  81;
  9: 0,  0,  0,   0,  50,  900, 2450, 2240, 810, 100;

%F Formula

The row polynomials satisfy (n^3+n^2-n^2*x)*P(n,x) = -(2*n^3-2*n^2-4*n-(5*n^2+3*n-2)*x+(2*n+2)*x^2)*P(n-1,x) + ((4*n^2+4*n)*x+(n^2-3*n-4)*x^2-(n+1)*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 - 2*n^2 - 4*n - (5*n^2 + 3*n - 2)*x + (2*n + 2)*x^2)*P[n - 1, x] + ((4*n^2 + 4*n)*x + (n^2 - 3*n - 4)*x^2 - (n + 1)*x^3)*D[P[n - 1, x], x])/(n^3 + 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 - 2*n^2 - 4*n - (5*n^2 + 3*n - 2)*x + (2*n + 2)*x^2)*v[i - 1] + ((4*n^2 + 4*n)*x + (n^2 - 3*n - 4)*x^2 - (n + 1)*x^3)*deriv(v[i - 1], x))/(n^3 + 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-d87fb5588738167f
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-d87fb5588738167f
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-d87fb5588738167f

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-d87fb5588738167f

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Riordan array (1+2x,x(1+x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - basis_map A111125 e-7b745496cf172799
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
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    no recorded property covers this requirement
  - theorem_check ma_wang_interval e-5a8f629e61d609b9 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
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    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
    
    preserver_image affine, shift 0 e-e470a688ac88b1cf
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
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    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
    
    preserver_image affine, shift 0 e-defa6799aaae8d2a
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038327 e-253047f151171d84
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
    
    preserver_image affine, shift 0 e-f8dba443f4d27d49
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038279 e-4738d2d3e61fb87a
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
    
    preserver_image affine, shift 0 e-fbfad365d270ad64
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A027467 e-71209083fd4bb877
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
    
    preserver_image affine, shift 0 e-1b92f65567f189e7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A007318 e-719615516772b132
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
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    preserver_image affine, shift 0 e-fa99d43ba99a1c05
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038243 e-94eaee4d49f2668d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
    
    preserver_image affine, shift 0 e-32ad399df63f844d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A027466 e-9b73612a2d328481
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-921246387fcf3a1f
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
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    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
    
    preserver_image affine, shift 0 e-85e6814e89c84b8b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
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    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
    
    preserver_image affine, shift 0 e-e475554f8902db13
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038291 e-e67dd89b1bcbf6c7
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
    
    preserver_image affine, shift 0 e-77c8319978ad9839
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A027465 e-f9e5e2bc826d3dc0
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
    
    preserver_image affine, shift 0 e-f73e26dac50fbf3d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)

Related sequences ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

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.