Real-rooted atlas

A038279 → 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
\(P_{n}(x) = \left(9 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19312
281549144
3729729243271728
46561874843749728120736
5590499841565610218703645243248832
6531441106288288573539366098415131227292985984
7478296911160261111602616200145206671541334345927218735831808
8430467211147912561339231328928208837200870992023216533721574646561429981696
9387420489116226146715496819561205308188602654094200884698446410446377292531441196835159780352
1034867844011162261467017433922005154968195609039811410361592456410044234901913187602391484517714705904961917364224
113138105960911506388523319177314205519177314205512784876137059662755306198875851024735139310789189885876877655845851177147743008370688
122824295364811129718145924207114993419423012777046601725958278495920511081864357976531836102279009096213081268953156759540315675954191318765314418916100448256
13254186582832911014751922759220295038455182692494914452222437457620435134624745722615983322032116199444067737249861016934392335216545123113622061119214746621785971594323106993205379072
1422876792454961106758364789818231309790377939308413053837252282711966017481188474644011654942373220058273589993219269610470813556203232684745693438780790948947007019332391725161120088469847829691283918464548864

Generated b-files: A001021 (row sums), A038227 (reversed triangle), A038293 ?

Computed in your browser ?

Technical details
Node id
n-7a4a43fcd9fa4a51 · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f9762973ec1089d7

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:         9,          3;
  2:        81,         54,          9;
  3:       729,        729,        243,         27;
  4:      6561,       8748,       4374,        972,        81;
  5:     59049,      98415,      65610,      21870,      3645,       243;
  6:    531441,    1062882,     885735,     393660,     98415,     13122,      729;
  7:   4782969,   11160261,   11160261,    6200145,   2066715,    413343,    45927,    2187;
  8:  43046721,  114791256,  133923132,   89282088,  37200870,   9920232,  1653372,  157464,   6561;
  9: 387420489, 1162261467, 1549681956, 1205308188, 602654094, 200884698, 44641044, 6377292, 531441, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (9+3*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[(9 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (9 + 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.

A038227: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,      9;
  2:     9,     54,      81;
  3:    27,    243,     729,      729;
  4:    81,    972,    4374,     8748,      6561;
  5:   243,   3645,   21870,    65610,     98415,     59049;
  6:   729,  13122,   98415,   393660,    885735,   1062882,     531441;
  7:  2187,  45927,  413343,  2066715,   6200145,  11160261,   11160261,    4782969;
  8:  6561, 157464, 1653372,  9920232,  37200870,  89282088,  133923132,  114791256,   43046721;
  9: 19683, 531441, 6377292, 44641044, 200884698, 602654094, 1205308188, 1549681956, 1162261467, 387420489;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (9+3*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[(9 + 3*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] = (9 + 3*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.

A038293: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:         1;
  1:         9,          3;
  2:        81,         54,          9;
  3:       729,        729,        243,         27;
  4:      6561,       8748,       4374,        972,        81;
  5:     59049,      98415,      65610,      21870,      3645,       243;
  6:    531441,    1062882,     885735,     393660,     98415,     13122,      729;
  7:   4782969,   11160261,   11160261,    6200145,   2066715,    413343,    45927,    2187;
  8:  43046721,  114791256,  133923132,   89282088,  37200870,   9920232,  1653372,  157464,   6561;
  9: 387420489, 1162261467, 1549681956, 1205308188, 602654094, 200884698, 44641044, 6377292, 531441, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (9+3*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[(9 + 3*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] = (9 + 3*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.

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-b4f43a8046f9cfa3
  • 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-b4f43a8046f9cfa3
  • 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-b4f43a8046f9cfa3

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 (16)

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