A038326: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.

OEIS matches ?

• A038337: reversed triangle, 30 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 ?
• A009967: 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 ?
• A038326: triangle, 36 terms agree, 8 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:         11,          12;
  2:        121,         264,          144;
  3:       1331,        4356,         4752,         1728;
  4:      14641,       63888,       104544,        76032,        20736;
  5:     161051,      878460,      1916640,      2090880,      1140480,       248832;
  6:    1771561,    11595672,     31624560,     45999360,     37635840,     16422912,      2985984;
  7:   19487171,   148811124,    487018224,    885487680,    965986560,    632282112,    229920768,     35831808;
  8:  214358881,  1870768416,   7142933952,  15584583168,  21251704320,  18546941952,  10116513792,   3153199104,   429981696;
  9: 2357947691, 23150759148, 101021494464, 257145622272, 420783745536, 459036813312, 333844955136, 156083355648, 42568187904, 5159780352;

The row polynomials satisfy P(n,x) = (11+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[(11 + 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] = (11 + 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:         11,          12;
  2:        121,         264,          144;
  3:       1331,        4356,         4752,         1728;
  4:      14641,       63888,       104544,        76032,        20736;
  5:     161051,      878460,      1916640,      2090880,      1140480,       248832;
  6:    1771561,    11595672,     31624560,     45999360,     37635840,     16422912,      2985984;
  7:   19487171,   148811124,    487018224,    885487680,    965986560,    632282112,    229920768,     35831808;
  8:  214358881,  1870768416,   7142933952,  15584583168,  21251704320,  18546941952,  10116513792,   3153199104,   429981696;
  9: 2357947691, 23150759148, 101021494464, 257145622272, 420783745536, 459036813312, 333844955136, 156083355648, 42568187904, 5159780352;

The row polynomials satisfy P(n,x) = (11+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[(11 + 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] = (11 + 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)

Triangle begins:
  0:          1;
  1:         12,          11;
  2:        144,         264,          121;
  3:       1728,        4752,         4356,         1331;
  4:      20736,       76032,       104544,        63888,        14641;
  5:     248832,     1140480,      2090880,      1916640,       878460,       161051;
  6:    2985984,    16422912,     37635840,     45999360,     31624560,     11595672,      1771561;
  7:   35831808,   229920768,    632282112,    965986560,    885487680,    487018224,    148811124,     19487171;
  8:  429981696,  3153199104,  10116513792,  18546941952,  21251704320,  15584583168,   7142933952,   1870768416,   214358881;
  9: 5159780352, 42568187904, 156083355648, 333844955136, 459036813312, 420783745536, 257145622272, 101021494464, 23150759148, 2357947691;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+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[(11 + 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] = (11 + 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)

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-da30941d57c5aa28
• 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-da30941d57c5aa28, e-f0c12876ab8620d1
• 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-da30941d57c5aa28, e-f0c12876ab8620d1
• 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-da30941d57c5aa28
• 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-f0c12876ab8620d1
• 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-c0fe49403ce57273
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 11] to roots in (-inf, 0]: an input root r <= 11 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-c0fe49403ce57273

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-c0fe49403ce57273 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-f0c12876ab8620d1 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-da30941d57c5aa28 root: theorem check

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

Used in 7 proof steps

• e-15af34f262d68ea6 basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
• e-a330da4062688974 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
• e-aadd11f319a04cdb basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
• e-c0f65529a8792e3f basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
• e-ffff6b452715bdc2 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
• e-4797824a301b09a2 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
• e-4797824a301b09a2 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.

Related sequences ?

Same fingerprint (3)

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

• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.
• A038281 → basis A013611
• A038314 → 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