A038327: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.

OEIS matches ?

• A013619: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A001022: 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 ?
• A038327: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · 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,          1;
  2:        144,         24,          1;
  3:       1728,        432,         36,         1;
  4:      20736,       6912,        864,        48,        1;
  5:     248832,     103680,      17280,      1440,       60,       1;
  6:    2985984,    1492992,     311040,     34560,     2160,      72,      1;
  7:   35831808,   20901888,    5225472,    725760,    60480,    3024,     84,    1;
  8:  429981696,  286654464,   83607552,  13934592,  1451520,   96768,   4032,   96,   1;
  9: 5159780352, 3869835264, 1289945088, 250822656, 31352832, 2612736, 145152, 5184, 108, 1;

The row polynomials satisfy P(n,x) = (12+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 + 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 + 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: 1,  12;
  2: 1,  24,  144;
  3: 1,  36,  432,   1728;
  4: 1,  48,  864,   6912,   20736;
  5: 1,  60, 1440,  17280,  103680,   248832;
  6: 1,  72, 2160,  34560,  311040,  1492992,   2985984;
  7: 1,  84, 3024,  60480,  725760,  5225472,  20901888,   35831808;
  8: 1,  96, 4032,  96768, 1451520, 13934592,  83607552,  286654464,  429981696;
  9: 1, 108, 5184, 145152, 2612736, 31352832, 250822656, 1289945088, 3869835264, 5159780352;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (12+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 + 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 + 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,          1;
  2:        144,         24,          1;
  3:       1728,        432,         36,         1;
  4:      20736,       6912,        864,        48,        1;
  5:     248832,     103680,      17280,      1440,       60,       1;
  6:    2985984,    1492992,     311040,     34560,     2160,      72,      1;
  7:   35831808,   20901888,    5225472,    725760,    60480,    3024,     84,    1;
  8:  429981696,  286654464,   83607552,  13934592,  1451520,   96768,   4032,   96,   1;
  9: 5159780352, 3869835264, 1289945088, 250822656, 31352832, 2612736, 145152, 5184, 108, 1;

The row polynomials satisfy P(n,x) = (12+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 + 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 + 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-cf37a3b130851ac7
• 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-75dac0c4b2fbfacb, e-cf37a3b130851ac7
• 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-75dac0c4b2fbfacb, e-cf37a3b130851ac7
• 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-cf37a3b130851ac7
• 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-75dac0c4b2fbfacb
• 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-9171fe67a9bd7e5c
• 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-9171fe67a9bd7e5c

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-9171fe67a9bd7e5c root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-75dac0c4b2fbfacb root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-cf37a3b130851ac7 root: theorem check

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

Used in 7 proof steps

• e-00e3cfe53baaea9b basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
• e-04b842e58817f21f basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
• e-950a320de9db6fb9 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
• e-d06691736a5246b1 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
• e-ec56a67a9fb88114 basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
• e-ea24c5115718581a obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
• e-ea24c5115718581a obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.

Related sequences ?

No candidate equalities or related sequences.

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