A139548 → 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

\(P_{n}(x) = \left(8 + 32\,x + 32\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	8	32	32													72
2	64	512	1536	2048	1024											5184
3	512	6144	30720	81920	122880	98304	32768									373248
4	4096	65536	458752	1835008	4587520	7340032	7340032	4194304	1048576							26873856
5	32768	655360	5898240	31457280	110100480	264241152	440401920	503316480	377487360	167772160	33554432					1934917632
6	262144	6291456	69206016	461373440	2076180480	6643777536	15502147584	26575110144	33218887680	29527900160	17716740096	6442450944	1073741824			139314069504
7	2097152	58720256	763363328	6106906624	33587986432	134351945728	403055837184	921270484992	1612223348736	2149631131648	2149631131648	1563368095744	781684047872	240518168576	34359738368	10030613004288

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-f8ceb867f43c1485 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-90334921cba1a1bb

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

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:         8,         32,          32;
  2:        64,        512,        1536,         2048,          1024;
  3:       512,       6144,       30720,        81920,        122880,          98304,           32768;
  4:      4096,      65536,      458752,      1835008,       4587520,        7340032,         7340032,         4194304,          1048576;
  5:     32768,     655360,     5898240,     31457280,     110100480,      264241152,       440401920,       503316480,        377487360,        167772160,         33554432;
  6:    262144,    6291456,    69206016,    461373440,    2076180480,     6643777536,     15502147584,     26575110144,      33218887680,      29527900160,      17716740096,       6442450944,        1073741824;
  7:   2097152,   58720256,   763363328,   6106906624,   33587986432,   134351945728,    403055837184,    921270484992,    1612223348736,    2149631131648,    2149631131648,    1563368095744,      781684047872,     240518168576,      34359738368;
  8:  16777216,  536870912,  8053063680,  75161927680,  488552529920,  2345052143616,   8598524526592,  24567212933120,   55276229099520,   98268851732480,  137576392425472,  150083337191424,   125069447659520,   76965813944320,   32985348833280,    8796093022208,    1099511627776;
  9: 134217728, 4831838208, 82141249536, 876173328384, 6571299962880, 36799279792128, 159463545765888, 546732156911616, 1503513431506944, 3341140958904320, 6014053726027776, 8747714510585856, 10205666929016832, 9420615626784768, 6729011161989120, 3588805953060864, 1345802232397824, 316659348799488, 35184372088832;

%F Formula

The row polynomials satisfy P(n,x) = (8+32*x+32*x^2)*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[(8 + 32*x + 32*x^2)*P[n - 1, 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] = (8 + 32*x + 32*x^2)*v[i - 1]); 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-770b6439e58a100f
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-770b6439e58a100f
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-770b6439e58a100f

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-770b6439e58a100f

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-4755dd2fd7289161 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (5)

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