Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.

OEIS matches ?

• A001025: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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.

Triangle begins:
  0:          144,            96,            16;
  1:         1728,          1728,           576,            64;
  2:        20736,         27648,         13824,          3072,           256;
  3:       248832,        414720,        276480,         92160,         15360,          1024;
  4:      2985984,       5971968,       4976640,       2211840,        552960,         73728,         4096;
  5:     35831808,      83607552,      83607552,      46448640,      15482880,       3096576,       344064,        16384;
  6:    429981696,    1146617856,    1337720832,     891813888,     371589120,      99090432,     16515072,      1572864,       65536;
  7:   5159780352,   15479341056,   20639121408,   16052649984,    8026324992,    2675441664,    594542592,     84934656,     7077888,     262144;
  8:  61917364224,  206391214080,  309586821120,  275188285440,  160526499840,   64210599936,  17836277760,   3397386240,   424673280,   31457280,   1048576;
  9: 743008370688, 2724364025856, 4540606709760, 4540606709760, 3027071139840, 1412633198592, 470877732864, 112113745920, 18685624320, 2076180480, 138412032, 4194304;

The row polynomials satisfy P(n,x) = (12+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 96*x + 16*x^2.

P[0, x_] := 144 + 96*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 96*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 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)

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-016885d52a801213
• 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-016885d52a801213, e-0a35d6a32dc0856a
• 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-016885d52a801213, e-0a35d6a32dc0856a
• 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-016885d52a801213
• 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-0a35d6a32dc0856a
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-19cc8e386ff5c847
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 12] to roots in (-inf, 0]: an input root r <= 12 gives the root (r - a)/d <= 0; a larger root gives a positive root
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-19cc8e386ff5c847

Proof chains

3 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

• basis_map_theorem e-19cc8e386ff5c847 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-0a35d6a32dc0856a root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-016885d52a801213 root: theorem check

  Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 7 proof steps

• e-1e482bc7f83a739a basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• e-6efc8fa5932928ce basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• e-77aaa200c8a95097 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• e-7b68ca495e98f7ea basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• e-dab838a9194ad2c3 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• e-60ab5e907432f9de obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• e-60ab5e907432f9de obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.

Related sequences ?

Same fingerprint (16)

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

• A130123 → basis A013610
• A013610 → D
• basis-map coefficients of A026729 → basis A013610
• basis-map coefficients of A021012 → basis A013610
• A038279 → basis A013610
• A038243 → basis A013609
• A027465: Cube of lower triangular normalized binomial matrix.
• A120909 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013610
• A013610 → basis A013611
• A120910 → basis A007318
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• basis-map coefficients of A030528, shifted → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610
• A136572 → basis A013610
• A027465 → D

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