basis-map coefficients of A112227 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence

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

\(P_{1}(x) = 1 + 8\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	1	8	16															25
2	1	12	48	64														125
3	1	16	96	256	256													625
4	1	20	160	640	1280	1024												3125
5	1	24	240	1280	3840	6144	4096											15625
6	1	28	336	2240	8960	21504	28672	16384										78125
7	1	32	448	3584	17920	57344	114688	131072	65536									390625
8	1	36	576	5376	32256	129024	344064	589824	589824	262144								1953125
9	1	40	720	7680	53760	258048	860160	1966080	2949120	2621440	1048576							9765625
10	1	44	880	10560	84480	473088	1892352	5406720	10813440	14417920	11534336	4194304						48828125
11	1	48	1056	14080	126720	811008	3784704	12976128	32440320	57671680	69206016	50331648	16777216					244140625
12	1	52	1248	18304	183040	1317888	7028736	28114944	84344832	187432960	299892736	327155712	218103808	67108864				1220703125
13	1	56	1456	23296	256256	2050048	12300288	56229888	196804608	524812288	1049624576	1526726656	1526726656	939524096	268435456			6103515625
14	1	60	1680	29120	349440	3075072	20500480	105431040	421724160	1312030720	3148873728	5725224960	7633633280	7046430720	4026531840	1073741824		30517578125
15	1	64	1920	35840	465920	4472832	32800768	187432960	843448320	2998927360	8396996608	18320719872	30534533120	37580963840	32212254720	17179869184	4294967296	152587890625

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Technical details

Node id: n-146f1cd501c36740 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-03d7b8417365c509

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 1)
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): yes

OEIS matches ?

A000351: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A074701: 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1,  8,  16;
   2: 1, 12,  48,    64;
   3: 1, 16,  96,   256,   256;
   4: 1, 20, 160,   640,  1280,   1024;
   5: 1, 24, 240,  1280,  3840,   6144,    4096;
   6: 1, 28, 336,  2240,  8960,  21504,   28672,   16384;
   7: 1, 32, 448,  3584, 17920,  57344,  114688,  131072,    65536;
   8: 1, 36, 576,  5376, 32256, 129024,  344064,  589824,   589824,   262144;
   9: 1, 40, 720,  7680, 53760, 258048,  860160, 1966080,  2949120,  2621440,  1048576;
  10: 1, 44, 880, 10560, 84480, 473088, 1892352, 5406720, 10813440, 14417920, 11534336, 4194304;

%F Formula

The row polynomials satisfy P(n,x) = (1+4*x)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 8*x + 16*x^2.

%t Mathematica

P[1, x_] := 1 + 8*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 16*x^2; for(n = 2, N, my(i = n); v[i] = (1 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-b91bf4c52fd06745
Coefficients of one sign. Every nonzero row, n >= 1: 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-b91bf4c52fd06745
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-b91bf4c52fd06745

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

basis_map A013611 e-b91bf4c52fd06745

Geometric basis map (elementary) polytool ?
- premise (sequence): basis-map coefficients of A scaled Hermite triangle.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-470643e70af29063 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - preserver_image mobius, shift 0 e-f9cce4eb6656235e
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A scaled Hermite triangle.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-470643e70af29063 (expanded above)

Related sequences ?

Same fingerprint (14)

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