A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.

OEIS matches ?

• A009965: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A038335: triangle, 31 terms agree, 7 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A038302: reversed triangle, 31 terms agree, 7 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?

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.

Triangle begins:
  0:          1;
  1:         12,           9;
  2:        144,         216,           81;
  3:       1728,        3888,         2916,          729;
  4:      20736,       62208,        69984,        34992,         6561;
  5:     248832,      933120,      1399680,      1049760,       393660,        59049;
  6:    2985984,    13436928,     25194240,     25194240,     14171760,      4251528,      531441;
  7:   35831808,   188116992,    423263232,    529079040,    396809280,    178564176,    44641044,     4782969;
  8:  429981696,  2579890176,   6772211712,  10158317568,   9523422720,   5714053632,  2142770112,   459165024,   43046721;
  9: 5159780352, 34828517376, 104485552128, 182849716224, 205705930752, 154279448064, 77139724032, 24794911296, 4649045868, 387420489;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (12 + 9*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)

Triangle begins:
  0:         1;
  1:         9,         12;
  2:        81,        216,         144;
  3:       729,       2916,        3888,        1728;
  4:      6561,      34992,       69984,       62208,        20736;
  5:     59049,     393660,     1049760,     1399680,       933120,       248832;
  6:    531441,    4251528,    14171760,    25194240,     25194240,     13436928,      2985984;
  7:   4782969,   44641044,   178564176,   396809280,    529079040,    423263232,    188116992,     35831808;
  8:  43046721,  459165024,  2142770112,  5714053632,   9523422720,  10158317568,   6772211712,   2579890176,   429981696;
  9: 387420489, 4649045868, 24794911296, 77139724032, 154279448064, 205705930752, 182849716224, 104485552128, 34828517376, 5159780352;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (12+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(12 + 9*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (12 + 9*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)

Triangle begins:
  0:          1;
  1:         12,           9;
  2:        144,         216,           81;
  3:       1728,        3888,         2916,          729;
  4:      20736,       62208,        69984,        34992,         6561;
  5:     248832,      933120,      1399680,      1049760,       393660,        59049;
  6:    2985984,    13436928,     25194240,     25194240,     14171760,      4251528,      531441;
  7:   35831808,   188116992,    423263232,    529079040,    396809280,    178564176,    44641044,     4782969;
  8:  429981696,  2579890176,   6772211712,  10158317568,   9523422720,   5714053632,  2142770112,   459165024,   43046721;
  9: 5159780352, 34828517376, 104485552128, 182849716224, 205705930752, 154279448064, 77139724032, 24794911296, 4649045868, 387420489;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(12 + 9*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (12 + 9*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)

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-03786f96284c3301
• 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-03786f96284c3301, e-2dcf274e8e5380c5
• 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-03786f96284c3301, e-2dcf274e8e5380c5
• 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-03786f96284c3301
• 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-2dcf274e8e5380c5
• 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-2d94fae2f976232e
• 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-2d94fae2f976232e

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-2d94fae2f976232e root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-2dcf274e8e5380c5 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-03786f96284c3301 root: theorem check

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

Used in 7 proof steps

• e-2be01244d609a451 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• e-3428d0ea6005b0ef basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• e-9b0235f0075e4320 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• e-b7b3e04e23eb02df basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• e-f8a7b8e38f7708ee basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• e-3fedd67ff8cfe3e1 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• e-3fedd67ff8cfe3e1 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.

Related sequences ?

Same fingerprint (6)

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

• A038330 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• A013609 → basis A013609
• A027465 → x ↦ x/(x+1)
• A013610 → basis A007318
• A120910 → basis A013611

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