A013620 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

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

\(P_{0}(x) = 3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	3	3								6
1	6	15	9							30
2	12	48	63	27						150
3	24	132	270	243	81					750
4	48	336	936	1296	891	243				3750
5	96	816	2880	5400	5670	3159	729			18750
6	192	1920	8208	19440	27540	23328	10935	2187		93750
7	384	4416	22176	63504	113400	129276	91854	37179	6561	468750

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-1ce1c7eadcd8d479 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-b821bbf6b18e27b2

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): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

A003948: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168875: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168923: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168827: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168683: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168731: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168779: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A167897: 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:    3,     3;
  1:    6,    15,      9;
  2:   12,    48,     63,     27;
  3:   24,   132,    270,    243,      81;
  4:   48,   336,    936,   1296,     891,     243;
  5:   96,   816,   2880,   5400,    5670,    3159,     729;
  6:  192,  1920,   8208,  19440,   27540,   23328,   10935,    2187;
  7:  384,  4416,  22176,  63504,  113400,  129276,   91854,   37179,    6561;
  8:  768,  9984,  57600, 193536,  417312,  598752,  571536,  349920,  124659,  19683;
  9: 1536, 22272, 145152, 559872, 1415232, 2449440, 2939328, 2414448, 1299078, 413343, 59049;

%F Formula

The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 3*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] = 3 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*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, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-cfb9e27fdcb193b1

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-cfb9e27fdcb193b1

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of coefficients in expansion of (2+3x)^n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-0ecfea69e82c3845 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-31e5a5a6b5727160 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - preserver_image reversal, shift 0 e-cb393d3715764915
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-1bc008755d2fc1a6 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    basis_map A007318 e-01e7c46c78122cbc
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+2*x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-a6866143cc223753 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    basis_map A013609 e-41b2234a78052e4f
    
    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)
    
    gamma_polynomial scale 1, shift 0 e-4c4f74d9ba733cff
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-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-eab26daf7024ea74 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image reversal, shift 0 e-6d15baf5e54583ee
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang e-0d085d05819c04f2 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    preserver_image reversal, shift 0 e-045f9d56b54ceaf2
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+2*x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-a6866143cc223753 (expanded above)
    
    ↵ basis_map A013609 e-41b2234a78052e4f (expanded above)
    
    ↵ gamma_polynomial scale 1, shift 0 e-4c4f74d9ba733cff (expanded above)
    
    parent_provers periodic_product e-d08b7f9df01d4f3c root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image mobius, shift 0 e-2c323256b27180b8
    
    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 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 (expanded above)
    
    ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (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)
    
    parent_provers periodic_product e-1697ba43e31b7729 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    basis_map A007318 e-4608a916413bc814
    
    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 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (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-17c52917511dc051
    
    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)
    
    ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (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)
    
    ↵ parent_provers periodic_product e-d08b7f9df01d4f3c (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-2c323256b27180b8 (expanded above)
    
    basis_map A013609 e-715b5f861d0b7004
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-0d085d05819c04f2 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-045f9d56b54ceaf2 (expanded above)
    
    ↵ parent_provers periodic_product e-1697ba43e31b7729 (expanded above)
    
    ↵ basis_map A007318 e-4608a916413bc814 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-17c52917511dc051 (expanded above)
    
    parent_provers periodic_product e-71e0b3707c4ee7db root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    preserver_image reversal, shift 0 e-dc7bfe74658d97ca
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (2+3x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-0ecfea69e82c3845 (expanded above)
    
    ↵ parent_provers periodic_product e-31e5a5a6b5727160 (expanded above)
    
    basis_map A013610 e-e9dcc990956c90e9
    
    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 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-033cf842aad3bdf7 (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 mobius, shift 0 e-57079e0e5fed1718
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-0d085d05819c04f2 (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-045f9d56b54ceaf2 (expanded above)
    
    ↵ parent_provers periodic_product e-1697ba43e31b7729 (expanded above)
    
    ↵ basis_map A007318 e-4608a916413bc814 (expanded above)
    
    ↵ preserver_image affine, shift 0 e-17c52917511dc051 (expanded above)
    
    preserver_image affine, shift 0 e-2582f481aa50840b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of coefficients in expansion of (1+2*x)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-a6866143cc223753 (expanded above)
    
    ↵ basis_map A013609 e-41b2234a78052e4f (expanded above)
    
    ↵ gamma_polynomial scale 1, shift 0 e-4c4f74d9ba733cff (expanded above)
    
    ↵ preserver_image reversal, shift 0 e-6d15baf5e54583ee (expanded above)
    
    ↵ parent_provers periodic_product e-d08b7f9df01d4f3c (expanded above)
    
    ↵ preserver_image mobius, shift 0 e-2c323256b27180b8 (expanded above)
  - ↵ basis_map A013610 e-e9dcc990956c90e9 (expanded above)
  - ↵ preserver_image mobius, shift 0 e-57079e0e5fed1718 (expanded above)
- premise (sequence): Triangle of coefficients in expansion of (2+3x)^n.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  no recorded property covers this requirement

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
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.