A023531 → basis A341287 → Pₙ + Pₙ₊₁

Proved for all n: all roots real what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	1	1															2
1	-1	2	1														2
2	-2	-1	4	1													2
3	1	-11	4	7	1												2
4	11	-18	-24	21	11	1											2
5	18	41	-129	-5	60	16	1										2
6	-41	317	-202	-384	155	134	22	1									2
7	-317	680	1065	-1974	-434	693	259	29	1								2
8	-680	-1767	8732	-3121	-7175	1477	2044	454	37	1							2
9	1767	-19911	25060	28069	-39206	-12054	10563	4926	741	46	1						2
10	19911	-68264	-53998	266091	-68485	-162854	16842	39117	10440	1145	56	1					2
11	68264	59643	-974533	1018215	806421	-983807	-335621	207141	112332	20185	1694	67	1				2
12	-59643	2076973	-4944068	-1145572	9162934	-2098888	-4447520	215710	924132	277057	36388	2419	79	1			2
13	-2076973	11905466	-4374447	-45032520	46000604	25353614	-30293978	-10330606	5115253	3053765	614328	62049	3354	92	1		2
14	-11905466	18577387	138254132	-323474423	12201826	354532542	-84096012	-143530244	3107533	26455858	8514506	1256619	101101	4536	106	1	2

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Technical details

Node id: n-24c1a22c5b1801ba · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-4c22959b3ea620d2

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
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;
  1:   -1,      2,     1;
  2:   -2,     -1,     4,     1;
  3:    1,    -11,     4,     7,      1;
  4:   11,    -18,   -24,    21,     11,      1;
  5:   18,     41,  -129,    -5,     60,     16,     1;
  6:  -41,    317,  -202,  -384,    155,    134,    22,    1;
  7: -317,    680,  1065, -1974,   -434,    693,   259,   29,   1;
  8: -680,  -1767,  8732, -3121,  -7175,   1477,  2044,  454,  37,  1;
  9: 1767, -19911, 25060, 28069, -39206, -12054, 10563, 4926, 741, 46, 1;

%F Formula

The row polynomials satisfy P(n,x) = x*P(n-1,x) - (1-x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + x.

%t Mathematica

P[0, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - (1 - 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 + x; for(n = 1, N, my(i = n + 1); v[i] = x*v[i - 1] - (1 - 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, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-cb47efc3e6c3b2c6

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

obreschkoff_combination e-cb47efc3e6c3b2c6

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - basis_map A341287 e-0f172f031b36afc5
    
    Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d
    
    Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
    
    premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-1991a148b9538b93 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-12303ad0e0b9f8eb
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
    
    parent_provers periodic_product e-b7eca039219a7f53 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-7e44b66e04c497d6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
  - preserver_image affine, shift 1 e-73143461db750abd
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-995393ce4e49a8d6 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    basis_map A008277 e-c52960bc5e97caa0
    
    Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-6e85c0971e84e272 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-ad67990681a86223
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-6e85c0971e84e272 (expanded above)
    
    preserver_image reversal, shift 0 e-d527982dc7c0aa3b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-8e580efc86a38a12 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-3256729839b26765
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-995393ce4e49a8d6 (expanded above)
    
    ↵ basis_map A008277 e-c52960bc5e97caa0 (expanded above)
- premise (sequence): Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k…
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  - ↵ preserver_image affine, shift 1 e-73143461db750abd (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
[Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[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
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.