Real-rooted atlas

A013620 → 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
\(P_{n}(x) = \left(5 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
151217
225120144289
3125900216017284913
4625600021600345602073683521
53125375001800004320005184002488321419857
6156252250001350000432000077760007464960298598424137569
77812513125009450000378000009072000013063680010450944035831808410338673
83906257500000630000003024000009072000001741824000209018880014332723204299816966975757441
919531254218750040500000022680000008164800000195955200003135283200032248627200193491763205159780352118587876497
10976562523437500025312500001620000000068040000000195955200000391910400000537477120000483729408000257989017600619173642242015993900449
114882812512890625001546875000011137500000053460000000017962560000004311014400000739031040000088683724800007094697984000340545503232074300837068834271896307633
122441406257031250000928125000007425000000004009500000000153964800000004311014400000088683724800000133025587200000141893959680000102163650969600445805022412808916100448256582622237229761
1312207031253808593750054843750000048262500000002895750000000012509640000000040030848000000096074035200000017293326336000002305776844800000221354577100800014488663228416005795465291366401069932053790729904578032905937
14610351562520507812500031992187500003071250000000020270250000000097297200000000035026992000000009607403520000000201755473920000003228087582720000038737050992640000338068808663040002028412851978240074895243765350401283918464548864168377826559400929

Generated b-files: A001026 (row sums), A038254, A038331 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-7b068e1328e63a95 · revision 1 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f055f948fcf55643

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:       5,       12;
  2:      25,      120,       144;
  3:     125,      900,      2160,       1728;
  4:     625,     6000,     21600,      34560,      20736;
  5:    3125,    37500,    180000,     432000,     518400,      248832;
  6:   15625,   225000,   1350000,    4320000,    7776000,     7464960,     2985984;
  7:   78125,  1312500,   9450000,   37800000,   90720000,   130636800,   104509440,    35831808;
  8:  390625,  7500000,  63000000,  302400000,  907200000,  1741824000,  2090188800,  1433272320,   429981696;
  9: 1953125, 42187500, 405000000, 2268000000, 8164800000, 19595520000, 31352832000, 32248627200, 19349176320, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (5+12*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[(5 + 12*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] = (5 + 12*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.

A038254: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,       12;
  2:      25,      120,       144;
  3:     125,      900,      2160,       1728;
  4:     625,     6000,     21600,      34560,      20736;
  5:    3125,    37500,    180000,     432000,     518400,      248832;
  6:   15625,   225000,   1350000,    4320000,    7776000,     7464960,     2985984;
  7:   78125,  1312500,   9450000,   37800000,   90720000,   130636800,   104509440,    35831808;
  8:  390625,  7500000,  63000000,  302400000,  907200000,  1741824000,  2090188800,  1433272320,   429981696;
  9: 1953125, 42187500, 405000000, 2268000000, 8164800000, 19595520000, 31352832000, 32248627200, 19349176320, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (5+12*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[(5 + 12*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] = (5 + 12*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.

A038331: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         12,           5;
  2:        144,         120,          25;
  3:       1728,        2160,         900,         125;
  4:      20736,       34560,       21600,        6000,         625;
  5:     248832,      518400,      432000,      180000,       37500,       3125;
  6:    2985984,     7464960,     7776000,     4320000,     1350000,     225000,      15625;
  7:   35831808,   104509440,   130636800,    90720000,    37800000,    9450000,    1312500,     78125;
  8:  429981696,  1433272320,  2090188800,  1741824000,   907200000,  302400000,   63000000,   7500000,   390625;
  9: 5159780352, 19349176320, 32248627200, 31352832000, 19595520000, 8164800000, 2268000000, 405000000, 42187500, 1953125;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+12*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[(5 + 12*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] = (5 + 12*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-1a3f71d8de8c31ec, e-545da5159276fba2
  • 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-1a3f71d8de8c31ec, e-545da5159276fba2
  • 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-1a3f71d8de8c31ec, e-545da5159276fba2

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (2)

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