A038324 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(21 + 10\,x\right)\,P_{n-1}(x) + \left(10\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	21	10							31
2	441	520	100						1061
3	9261	20530	9300	1000					40091
4	194481	729040	586600	144000	10000				1664121
5	4084101	24545050	31341000	13210000	2050000	100000			75330151
6	85766121	801737560	1530431500	987120000	257150000	27600000	1000000		3690805181
7	1801088541	25711525570	70765067100	65647435000	25557350000	4531100000	357000000	10000000	194380566211

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-e3c4685fe4960b3e · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-4c7c5e901d4aacf3

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): no

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:           21,             10;
  2:          441,            520,             100;
  3:         9261,          20530,            9300,            1000;
  4:       194481,         729040,          586600,          144000,           10000;
  5:      4084101,       24545050,        31341000,        13210000,         2050000,         100000;
  6:     85766121,      801737560,      1530431500,       987120000,       257150000,       27600000,        1000000;
  7:   1801088541,    25711525570,     70765067100,     65647435000,     25557350000,     4531100000,      357000000,      10000000;
  8:  37822859361,   815068178080,   3158483006800,   4055669856000,   2215472700000,   577281600000,    74228000000,    4480000000,   100000000;
  9: 794280046581, 25645342114090, 137648485059600, 238423992724000, 175700533260000, 63141720600000, 11785284000000, 1149960000000, 54900000000, 1000000000;

%F Formula

The row polynomials satisfy P(n,x) = (21+10*x)*P(n-1,x) + 10*x*d/dx 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[(21 + 10*x)*P[n - 1, x] + 10*x*D[P[n - 1, x], 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] = (21 + 10*x)*v[i - 1] + 10*x*deriv(v[i - 1], x)); 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-6a3d937c421a3f39
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-6a3d937c421a3f39
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-6a3d937c421a3f39

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 A008277 e-6a3d937c421a3f39

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-0ddfaa48a1d96c8a root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - basis_map A013609 e-0f31cc897fc45a50
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-0a7984b7b9125974 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    parent_provers periodic_product e-7949976749bfedcf root: theorem check
    
    Parent-row provers of polytool polytool ?
  - parent_provers periodic_product e-96e2fb6c9c1a89f9 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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