Real-rooted atlas

A038291 → 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
\(P_{n}(x) = \left(10 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110414
21008016196
310001200480642744
410000160009600256025638416
510000020000016000064000128001024537824
610000002400000240000012800003840006144040967529536
7100000002800000033600000224000008960000215040028672016384105413504
810000000032000000044800000035840000017920000057344000114688001310720655361475789056
910000000003600000000576000000053760000003225600000129024000034406400058982400589824026214420661046784
1010000000000400000000007200000000076800000000537600000002580480000086016000001966080000294912000262144001048576289254654976
1110000000000044000000000088000000000010560000000008448000000004730880000001892352000005406720000010813440000144179200011534336041943044049565169664
12100000000000048000000000001056000000000014080000000000126720000000008110080000000378470400000012976128000003244032000005767168000069206016005033164801677721656693912375296
13100000000000005200000000000012480000000000018304000000000018304000000000013178880000000070287360000000281149440000008434483200000187432960000029989273600032715571200218103808067108864793714773254144
141000000000000005600000000000001456000000000000232960000000000025625600000000002050048000000000123002880000000056229888000000019680460800000052481228800000104962457600001526726656000152672665600939524096026843545611112006825558016

Generated b-files: A001023 (row sums), A038240 (reversed triangle), A038306 ?

Computed in your browser ?

Technical details
Node id
n-8afa7b2308002172 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-9629d1e9c04dbfbe

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:         10,          4;
  2:        100,         80,         16;
  3:       1000,       1200,        480,         64;
  4:      10000,      16000,       9600,       2560,        256;
  5:     100000,     200000,     160000,      64000,      12800,       1024;
  6:    1000000,    2400000,    2400000,    1280000,     384000,      61440,      4096;
  7:   10000000,   28000000,   33600000,   22400000,    8960000,    2150400,    286720,    16384;
  8:  100000000,  320000000,  448000000,  358400000,  179200000,   57344000,  11468800,  1310720,   65536;
  9: 1000000000, 3600000000, 5760000000, 5376000000, 3225600000, 1290240000, 344064000, 58982400, 5898240, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (10+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[(10 + 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] = (10 + 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.

A038240: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,      10;
  2:     16,      80,      100;
  3:     64,     480,     1200,      1000;
  4:    256,    2560,     9600,     16000,      10000;
  5:   1024,   12800,    64000,    160000,     200000,     100000;
  6:   4096,   61440,   384000,   1280000,    2400000,    2400000,    1000000;
  7:  16384,  286720,  2150400,   8960000,   22400000,   33600000,   28000000,   10000000;
  8:  65536, 1310720, 11468800,  57344000,  179200000,  358400000,  448000000,  320000000,  100000000;
  9: 262144, 5898240, 58982400, 344064000, 1290240000, 3225600000, 5376000000, 5760000000, 3600000000, 1000000000;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (10+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[(10 + 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] = (10 + 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.

A038306: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         10,          4;
  2:        100,         80,         16;
  3:       1000,       1200,        480,         64;
  4:      10000,      16000,       9600,       2560,        256;
  5:     100000,     200000,     160000,      64000,      12800,       1024;
  6:    1000000,    2400000,    2400000,    1280000,     384000,      61440,      4096;
  7:   10000000,   28000000,   33600000,   22400000,    8960000,    2150400,    286720,    16384;
  8:  100000000,  320000000,  448000000,  358400000,  179200000,   57344000,  11468800,  1310720,   65536;
  9: 1000000000, 3600000000, 5760000000, 5376000000, 3225600000, 1290240000, 344064000, 58982400, 5898240, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (10+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[(10 + 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] = (10 + 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-8c5f302cdd82df95
  • 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-8c5f302cdd82df95
  • 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-8c5f302cdd82df95

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 (6)

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