A137320 → basis A013610

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 6\,x + n\right)\,P_{n-1}(x)\)

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

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	6														8
2	6	30	36													72
3	24	156	324	216												720
4	120	924	2556	3024	1296											7920
5	720	6264	20880	33480	25920	7776										95040
6	5040	48168	183744	359640	382320	209952	46656									1235520
7	40320	415584	1758960	3979584	5216400	3973536	1632960	279936								17297280
8	362880	3982176	18324144	46370016	70825104	67060224	38537856	12317184	1679616							259459200
9	3628800	41999040	207134496	573645024	986471136	1095552864	787739904	354398976	90699264	10077696						4151347200
10	39916800	483762240	2530473696	7552902240	14293052640	17969908320	15238456128	8624828160	3124085760	655050240	60466176					70572902400
11	479001600	6044647680	33268257792	105817669056	216834045120	301397215680	290680923456	194928674688	89237998080	26605117440	4655895552	362797056				1270312243200
12	6227020800	81454429440	468755237376	1575239244480	3453748600896	5219168074560	5587235299008	4278158311680	2329666023168	881294515200	220157346816	32651735040	2176782336			24135932620800
13	87178291200	1177724136960	7051299899904	24865880846976	57803915879424	93790844649216	109536302633472	93417628157568	58284274194432	26316119351808	8369969946624	1778068371456	226385362944	13060694016		482718652416000
14	1307674368000	18188931801600	112835843320320	415296012104064	1016254023273216	1753686165014784	2205789607397376	2058482238164352	1434769881861888	744447435443712	283446265310208	76890845251584	14064190672896	1554222587904	78364164096	10137091700736000

Generated b-files: A049388 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-8a9c0145328414df · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-fe4d40ee62c150a8

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 ?

A049388: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,        6;
  2:       6,       30,        36;
  3:      24,      156,       324,       216;
  4:     120,      924,      2556,      3024,      1296;
  5:     720,     6264,     20880,     33480,     25920,       7776;
  6:    5040,    48168,    183744,    359640,    382320,     209952,     46656;
  7:   40320,   415584,   1758960,   3979584,   5216400,    3973536,   1632960,    279936;
  8:  362880,  3982176,  18324144,  46370016,  70825104,   67060224,  38537856,  12317184,  1679616;
  9: 3628800, 41999040, 207134496, 573645024, 986471136, 1095552864, 787739904, 354398976, 90699264, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+6*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 + 1 + 6*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 + 1 + 6*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-a84b2eeb296ce438
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-a84b2eeb296ce438
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-a84b2eeb296ce438

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 A013610 e-a84b2eeb296ce438

Geometric basis map (elementary) polytool ?
- premise (sequence): Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-6aa1e3cc7e408461 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.