Real-rooted atlas

A038303 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(11 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111415
21218816225
313311452528643375
4146412129611616281625650625
516105129282021296077440140801024759375
6177156138652243513840170368046464067584409611390625
71948717149603708541131363279584011925760260198431539216384170859375
821435888162358947279365932857720678426236672076324864138772481441792655362562890625
9235794769177169197161122461049695239119365194861056188904038445794918471368704648806426214438443359375
1025937424601943179076401543383943201496614732809523911936041558888448125936025602616852480356843520288358401048576576650390625
1128531167061111412466824442074993968080226362978336016462762060808381042503683047651819527915978752014392688640174456832012687769641943048649755859375
12313842837672113694960189328273899203786563319990348928027163557400320158042515783686704834002944208981839052847495872512076761006080837392793655364812816777216129746337890625
133452271214393116319827558949235606896492252847475861989670443159874536064028250099696332813697018034585649807338307584135838195384322744205967360399157231616395858411522399141888671088641946195068359375
143797498335832411933271880060136456955171650577666466206785538566646620678553856483390594803916826366759716577281095761442766848348651368153088845215437946881536755341721620320731791361847339253761033476505626843545629192926025390625

Generated b-files: A001024 (row sums), A038241 (reversed triangle), A038318 ?

Computed in your browser ?

Technical details
Node id
n-e03395d80bc5749d · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-432dc7cc45625337

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:         11,          4;
  2:        121,         88,          16;
  3:       1331,       1452,         528,         64;
  4:      14641,      21296,       11616,       2816,        256;
  5:     161051,     292820,      212960,      77440,      14080,       1024;
  6:    1771561,    3865224,     3513840,    1703680,     464640,      67584,      4096;
  7:   19487171,   49603708,    54113136,   32795840,   11925760,    2601984,    315392,    16384;
  8:  214358881,  623589472,   793659328,  577206784,  262366720,   76324864,  13877248,  1441792,   65536;
  9: 2357947691, 7716919716, 11224610496, 9523911936, 5194861056, 1889040384, 457949184, 71368704, 6488064, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (11+4*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[(11 + 4*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] = (11 + 4*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.

A038241: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,      11;
  2:     16,      88,      121;
  3:     64,     528,     1452,      1331;
  4:    256,    2816,    11616,     21296,      14641;
  5:   1024,   14080,    77440,    212960,     292820,     161051;
  6:   4096,   67584,   464640,   1703680,    3513840,    3865224,    1771561;
  7:  16384,  315392,  2601984,  11925760,   32795840,   54113136,   49603708,    19487171;
  8:  65536, 1441792, 13877248,  76324864,  262366720,  577206784,  793659328,   623589472,  214358881;
  9: 262144, 6488064, 71368704, 457949184, 1889040384, 5194861056, 9523911936, 11224610496, 7716919716, 2357947691;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+4*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[(11 + 4*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] = (11 + 4*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.

A038318: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         11,          4;
  2:        121,         88,          16;
  3:       1331,       1452,         528,         64;
  4:      14641,      21296,       11616,       2816,        256;
  5:     161051,     292820,      212960,      77440,      14080,       1024;
  6:    1771561,    3865224,     3513840,    1703680,     464640,      67584,      4096;
  7:   19487171,   49603708,    54113136,   32795840,   11925760,    2601984,    315392,    16384;
  8:  214358881,  623589472,   793659328,  577206784,  262366720,   76324864,  13877248,  1441792,   65536;
  9: 2357947691, 7716919716, 11224610496, 9523911936, 5194861056, 1889040384, 457949184, 71368704, 6488064, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (11+4*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[(11 + 4*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] = (11 + 4*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-551d764375152d3c
  • 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-551d764375152d3c
  • 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-551d764375152d3c

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