Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.

OEIS matches ?

• A001024: 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,            72,             9;
  1:         1728,          1296,           324,            27;
  2:        20736,         20736,          7776,          1296,           81;
  3:       248832,        311040,        155520,         38880,         4860,          243;
  4:      2985984,       4478976,       2799360,        933120,       174960,        17496,         729;
  5:     35831808,      62705664,      47029248,      19595520,      4898880,       734832,       61236,        2187;
  6:    429981696,     859963392,     752467968,     376233984,    117573120,     23514624,     2939328,      209952,       6561;
  7:   5159780352,   11609505792,   11609505792,    6772211712,   2539579392,    634894848,   105815808,    11337408,     708588,     19683;
  8:  61917364224,  154793410560,  174142586880,  116095057920,  50791587840,  15237476352,  3174474240,   453496320,   42515280,   2361960,   59049;
  9: 743008370688, 2043273019392, 2554091274240, 1915568455680, 957784227840, 335224479744, 83806119936, 14965378560, 1870672320, 155889360, 7794468, 177147;

The row polynomials satisfy P(n,x) = (12+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 72*x + 9*x^2.

P[0, x_] := 144 + 72*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 3*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 + 72*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 3*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-d1ea7de7e1f61f8b
• 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-a6c3b06cf2d0ed0d, e-d1ea7de7e1f61f8b
• 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-a6c3b06cf2d0ed0d, e-d1ea7de7e1f61f8b
• 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-d1ea7de7e1f61f8b
• 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-a6c3b06cf2d0ed0d
• 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-1f4aa302c408d4a8
• 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-1f4aa302c408d4a8

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-1f4aa302c408d4a8 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-a6c3b06cf2d0ed0d root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-d1ea7de7e1f61f8b root: theorem check

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

Used in 7 proof steps

• e-a5d7492b5780a0c3 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• e-a617f0e181e2ec6f basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• e-f14e261f691e9869 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• e-f3e6c91e6b1bdc2d basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• e-fe430a680aa07591 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• e-bebf949c785e986b obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• e-bebf949c785e986b obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.

Related sequences ?

Same fingerprint (14)

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

• basis-map coefficients of A030528, shifted → basis A013611
• basis-map coefficients of A021012 → basis A013611
• A038231 → D
• A136572 → basis A013611
• A027467 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013611
• basis-map coefficients of A026729 → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A013611 → D
• A038330 → basis A007318
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• basis-map coefficients of A112227 → basis A013611
• A130123 → basis A013611
• A027466 → basis A013609

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