A038208 → θ+1

nonnegative coefficients only rational roots

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

Recurrence

\(\left(1 + n x\right)\,P_{n}(x) = \left(2 + 4\,x + 2\,x^{2} + 2\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	4	16	12						32
3	8	48	72	32					160
4	16	128	288	256	80				768
5	32	320	960	1280	800	192			3584
6	64	768	2880	5120	4800	2304	448		16384
7	128	1792	8064	17920	22400	16128	6272	1024	73728

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-777049cbeb01df27 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-e99c3824c4864868

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 ?

A292044: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,    4;
  2:   4,   16,    12;
  3:   8,   48,    72,     32;
  4:  16,  128,   288,    256,     80;
  5:  32,  320,   960,   1280,    800,    192;
  6:  64,  768,  2880,   5120,   4800,   2304,    448;
  7: 128, 1792,  8064,  17920,  22400,  16128,   6272,   1024;
  8: 256, 4096, 21504,  57344,  89600,  86016,  50176,  16384,  2304;
  9: 512, 9216, 55296, 172032, 322560, 387072, 301056, 147456, 41472, 5120;

%F Formula

The row polynomials satisfy (1+n*x)*P(n,x) = (2+(2*n+4)*x+(2*n+2)*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[Together[((2 + (2*n + 4)*x + (2*n + 2)*x^2)*P[n - 1, x])/(1 + n*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] = ((2 + (2*n + 4)*x + (2*n + 2)*x^2)*v[i - 1])/(1 + n*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-0211c2ba010af36a
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-0211c2ba010af36a
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-0211c2ba010af36a
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-0211c2ba010af36a

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

preserver_image euler_polynomial, shift 0 e-0211c2ba010af36a

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-4f4e9585c5795aa9 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-0d998323de038265 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - basis_map A007318 e-1de26a40255316f7
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-7963c2f0e7b86ba8 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-37a63c273567eea5
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-7963c2f0e7b86ba8 (expanded above)
  - preserver_image reversal, shift 0 e-cf1dcb38aee4f2af
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-4f4e9585c5795aa9 (expanded above)
    
    ↵ parent_provers periodic_product e-0d998323de038265 (expanded above)
    
    ↵ basis_map A007318 e-1de26a40255316f7 (expanded above)
    
    basis_map A013609 e-e548dab430fb29bc
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): binomial
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-317eb4ccca60c6ac root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-033cf842aad3bdf7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): binomial
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
    
    basis_map A007318 e-095baa2a9dfe188f
    
    Geometric basis map (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
    
    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)
    
    parent_provers periodic_product e-13983862a411779c root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image reversal, shift 0 e-989c95e5c893fce9
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1-x,x/(1-x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-b62074ba45057e82 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    preserver_image reversal, shift 0 e-0e058c428dfdfeb6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 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-d82c9b48e4b31a6b root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)
    
    preserver_image affine, shift 0 e-3a52de7742ed9314
    
    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 mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
    
    ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
    
    ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
    
    ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
    
    ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
    
    preserver_image affine, shift 0 e-07e3ff75f72fdf91
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-7963c2f0e7b86ba8 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-37a63c273567eea5 (expanded above)
  - ↵ basis_map A013609 e-e548dab430fb29bc (expanded above)
  - ↵ preserver_image affine, shift 0 e-07e3ff75f72fdf91 (expanded above)

Related sequences ?

Same fingerprint (27)

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

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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.