Real-rooted atlas

A257613: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.

A257613 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A257613
\(P_{n}(x) = \left(4 + 2\,x + 2\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
216481680
36441641664960
425631366656313625613440
51024218888460884608218881024215040
64096145664939520169216093952014566440963870720
7163849395209555456281958402819584095554569395201638477414400
865536593203291475968415734784676700160415734784914759685932032655361703116800
9262144369029128385843205620963328139406888961394068889656209633288385843203690291226214440874803200
10104857622718464074467328007130415104025722368000039033928908825722368000071304151040744673280022718464010485761062744883200
1141943041388273664645719244808619761664004370158878720958032892723295803289272324370158878720861976166400645719244801388273664419430429756856729600
1216777216843869388854989396377610040344002560696814298726402127874647982083065705256714242127874647982086968142987264010040344002560549893963776843869388816777216892705701888000
13671088645110192537646185577512961136008951562241057064726528000437265310462771287353027771105288735302777110528437265310462771210570647265280001136008951562244618557751296511019253766710886428566582460416000
1426843545630862481817638379315920896125609145309593615411198202085376844725674484039682272179065263226883144708999759790082272179065263226888447256744840396815411198202085376125609145309593638379315920896308624818176268435456971263803654144000

Generated b-files: A051580 (row sums), A257613, A257613 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-c35e92273dac3e06 · revision 1 · JSON
Definition source
oeis_catalog:A257613
Fingerprint
f6-9f31c05ba8767dc5

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: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

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,        4;
  2:     16,       48,        16;
  3:     64,      416,       416,         64;
  4:    256,     3136,      6656,       3136,         256;
  5:   1024,    21888,     84608,      84608,       21888,        1024;
  6:   4096,   145664,    939520,    1692160,      939520,      145664,       4096;
  7:  16384,   939520,   9555456,   28195840,    28195840,     9555456,     939520,     16384;
  8:  65536,  5932032,  91475968,  415734784,   676700160,   415734784,   91475968,   5932032,    65536;
  9: 262144, 36902912, 838584320, 5620963328, 13940688896, 13940688896, 5620963328, 838584320, 36902912, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (4+(2*n+2)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx 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 + (2*n + 2)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], 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 + (2*n + 2)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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.

A257613: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,        4;
  2:     16,       48,        16;
  3:     64,      416,       416,         64;
  4:    256,     3136,      6656,       3136,         256;
  5:   1024,    21888,     84608,      84608,       21888,        1024;
  6:   4096,   145664,    939520,    1692160,      939520,      145664,       4096;
  7:  16384,   939520,   9555456,   28195840,    28195840,     9555456,     939520,     16384;
  8:  65536,  5932032,  91475968,  415734784,   676700160,   415734784,   91475968,   5932032,    65536;
  9: 262144, 36902912, 838584320, 5620963328, 13940688896, 13940688896, 5620963328, 838584320, 36902912, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (4+(2*n+2)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx 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 + (2*n + 2)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], 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 + (2*n + 2)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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.

A257613: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,        4;
  2:     16,       48,        16;
  3:     64,      416,       416,         64;
  4:    256,     3136,      6656,       3136,         256;
  5:   1024,    21888,     84608,      84608,       21888,        1024;
  6:   4096,   145664,    939520,    1692160,      939520,      145664,       4096;
  7:  16384,   939520,   9555456,   28195840,    28195840,     9555456,     939520,     16384;
  8:  65536,  5932032,  91475968,  415734784,   676700160,   415734784,   91475968,   5932032,    65536;
  9: 262144, 36902912, 838584320, 5620963328, 13940688896, 13940688896, 5620963328, 838584320, 36902912, 262144;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (4+(2*n+2)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx 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 + (2*n + 2)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], 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 + (2*n + 2)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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-5e8558eb5869f0da
  • 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-5e8558eb5869f0da
  • 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-5e8558eb5869f0da
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-5e8558eb5869f0da

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

Used in 3 proof steps

Same fingerprint (2)

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

References

  1. [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