A269947 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	2	6	4													12
3	18	58	48	8												132
4	504	1660	1460	320	16											3960
5	32760	108908	98220	23720	1680	32										265320
6	4127760	13787928	12593536	3185160	259120	7392	64									33960960
7	895723920	3000235896	2760373168	716366792	62599360	2122304	28672	128								7437450240
8	308129028480	1033872596064	955568841584	251950922784	22966913424	855271296	14107776	101376	256							2573357783040
9	158070191610240	530992899837792	492274560924720	131161961071360	12285928432080	484688001696	8947831680	80221440	334080	512						1325279258265600
10	115391239875475200	387940957264808640	360422415274721184	96732780703942240	9231051677561120	378394098102240	7501293129792	76457314560	404321280	1041920	1024					970104417050419200
11	115506631115350675200	388559680701824399040	361558719604525522464	97550358315195624608	9433748290646565600	397234595555464480	8265582619126272	91536358134144	557640230400	1851604480	3108864	2048				973014730301570457600
12	153854832645647099366400	517792507957060800871680	482373333874631644720128	130660194715049623022784	12760853439771616628416	547983977861171818560	11804225239787123264	138457594272932352	925849503161088	3581617628160	7844215808	8945664	4096			1298001650222294990438400
13	266015005644323834804505600	895570955923049418905867520	835059079285152235322844672	226876223330070061495833792	22324835986795224396576832	972986004601509307547072	21505473395314279760576	263001630977474283136	1877708979511385856	8044315885410816	20725884388352	31155484672	24973312	8192		2246840856534792628448870400
14	584700982406223788900303308800	1968996991130151270424705820160	1837250998180610712077424324096	500344057038064299638488364160	49523741945636043346667544320	2183272910087707906781617920	49215002532103805528840192	621088531679117033854080	4653207598920974677760	21436824275155745280	61644125656419328	109931524085760	117202309120	67952640	16384	4943049884376543782587514880000

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Technical details

Node id: n-7cfac8d939582bff · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-84bd2520c71a96df

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): 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:               1,               2;
  2:               2,               6,               4;
  3:              18,              58,              48,               8;
  4:             504,            1660,            1460,             320,             16;
  5:           32760,          108908,           98220,           23720,           1680,           32;
  6:         4127760,        13787928,        12593536,         3185160,         259120,         7392,         64;
  7:       895723920,      3000235896,      2760373168,       716366792,       62599360,      2122304,      28672,      128;
  8:    308129028480,   1033872596064,    955568841584,    251950922784,    22966913424,    855271296,   14107776,   101376,    256;
  9: 158070191610240, 530992899837792, 492274560924720, 131161961071360, 12285928432080, 484688001696, 8947831680, 80221440, 334080, 512;

%F Formula

The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n+2*x)*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[(n^3 - 3*n^2 + 3*n + 2*x)*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] = (n^3 - 3*n^2 + 3*n + 2*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 - 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-e62c7427c0d251dd
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-e62c7427c0d251dd
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-e62c7427c0d251dd

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 A013609 e-e62c7427c0d251dd

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-f7365f1dbad216ac root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.