A059473 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	4	2							6
3	8	8							16
4	16	24	4						44
5	32	64	24						120
6	64	160	96	8					328
7	128	384	320	64					896
8	256	896	960	320	16				2448
9	512	2048	2688	1280	160				6688
10	1024	4608	7168	4480	960	32			18272
11	2048	10240	18432	14336	4480	384			49920
12	4096	22528	46080	43008	17920	2688	64		136384
13	8192	49152	112640	122880	64512	14336	896		372608
14	16384	106496	270336	337920	215040	64512	7168	128	1017984

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

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

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

Technical details

Node id: n-a5662819bbc56ccb · revision 2 · JSON
Definition source: derived:gamma_polynomial
derived:reversal (equal by e-d6ef3d3814ca6f12)
Fingerprint: f6-b928c80a61c812bf

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 ?

A002605: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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;
  2:   4,    2;
  3:   8,    8;
  4:  16,   24,    4;
  5:  32,   64,   24;
  6:  64,  160,   96,    8;
  7: 128,  384,  320,   64;
  8: 256,  896,  960,  320,  16;
  9: 512, 2048, 2688, 1280, 160;

%F Formula

The row polynomials satisfy P(n,x) = 2*P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] + 2*x*P[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; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] + 2*x*v[i - 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)

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-9bce1221c4367de4, e-babfdb0b6fa40ea8
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-9bce1221c4367de4, e-babfdb0b6fa40ea8
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-9bce1221c4367de4, e-babfdb0b6fa40ea8

Proof chains

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

definition_equality e-d6ef3d3814ca6f12 root: theorem check

Ore-algebra identity of two definitions polytool ?
gamma_polynomial scale 1, shift 0 e-babfdb0b6fa40ea8

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-0ee228386582da48 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-c68c82cb1005adf9
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-0ee228386582da48 (expanded above)
preserver_image reversal, shift 0 e-9bce1221c4367de4

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 (-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 ?

Same fingerprint (1)

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

A099040: Riordan array (1, 2+2x).

References

[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
[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