Real-rooted atlas

A013610 → 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
\(P_{n}(x) = \left(4 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14913
2167281169
3644329727292197
42562304777611664656128561
51024115205184011664013122059049371293
6409655296311040933120157464014171765314414826809
71638425804817418246531840146966401984046414880348478296962748517
865536117964892897284180377611757312021163161623808556815305500843046721815730721
9262144530841647775744250822656846526464190468454428570268162754990144154968195638742048910604499373
101048576235929602388787201433272320564350976015237476352285702681603673320192030993639120154968195603486784401137858491849
1141943041038090241167851520788299776035473489920111741493248251418359808404065221120454573373760340930030320153418513644313810596091792160394037
121677721645298483256056872964204265472021284093952076622738227220113468784643879026122752545488048512054548804851203682044327456150629086123228242953648123298085122481
136710886419629342722649961267221862180454412297476505604980477984768149414339543043361822639718456730757045248709134463065606382210167590439163562392032146863358970122541865828329302875106592253
1426843545684557168641236648591361112983732224688658684313630989640794112104590037680128268945811177472529487065755648794230598633472893509423462656731053164651264411217405116336142344486386424228767924549613937376385699289

Generated b-files: A001022 (row sums), A038239, A038294 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-a313e9d783515c70 · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-12e1fe8c90bb4450

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 ?

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:      4,       9;
  2:     16,      72,       81;
  3:     64,     432,      972,       729;
  4:    256,    2304,     7776,     11664,      6561;
  5:   1024,   11520,    51840,    116640,    131220,      59049;
  6:   4096,   55296,   311040,    933120,   1574640,    1417176,     531441;
  7:  16384,  258048,  1741824,   6531840,  14696640,   19840464,   14880348,    4782969;
  8:  65536, 1179648,  9289728,  41803776, 117573120,  211631616,  238085568,  153055008,   43046721;
  9: 262144, 5308416, 47775744, 250822656, 846526464, 1904684544, 2857026816, 2754990144, 1549681956, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (4+9*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[(4 + 9*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] = (4 + 9*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.

A038239: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       9;
  2:     16,      72,       81;
  3:     64,     432,      972,       729;
  4:    256,    2304,     7776,     11664,      6561;
  5:   1024,   11520,    51840,    116640,    131220,      59049;
  6:   4096,   55296,   311040,    933120,   1574640,    1417176,     531441;
  7:  16384,  258048,  1741824,   6531840,  14696640,   19840464,   14880348,    4782969;
  8:  65536, 1179648,  9289728,  41803776, 117573120,  211631616,  238085568,  153055008,   43046721;
  9: 262144, 5308416, 47775744, 250822656, 846526464, 1904684544, 2857026816, 2754990144, 1549681956, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (4+9*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[(4 + 9*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (4 + 9*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038294: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:         1;
  1:         9,          4;
  2:        81,         72,         16;
  3:       729,        972,        432,         64;
  4:      6561,      11664,       7776,       2304,        256;
  5:     59049,     131220,     116640,      51840,      11520,      1024;
  6:    531441,    1417176,    1574640,     933120,     311040,     55296,      4096;
  7:   4782969,   14880348,   19840464,   14696640,    6531840,   1741824,    258048,    16384;
  8:  43046721,  153055008,  238085568,  211631616,  117573120,  41803776,   9289728,  1179648,   65536;
  9: 387420489, 1549681956, 2754990144, 2857026816, 1904684544, 846526464, 250822656, 47775744, 5308416, 262144;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (4+9*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[(4 + 9*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (4 + 9*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-39440d1ff583ab52
  • 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-39440d1ff583ab52
  • 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-39440d1ff583ab52

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

Same fingerprint (1)

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

References

  1. [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
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [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