A130123 → basis A013611

OEIS matches ?

• A155148: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A038280: reversed triangle, 39 terms agree, 8 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A011557: 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 ?
• A178500: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A178501: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A038214: triangle, 40 terms agree, 8 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:   2,     8;
  2:   4,    32,     64;
  3:   8,    96,    384,     512;
  4:  16,   256,   1536,    4096,     4096;
  5:  32,   640,   5120,   20480,    40960,    32768;
  6:  64,  1536,  15360,   81920,   245760,   393216,    262144;
  7: 128,  3584,  43008,  286720,  1146880,  2752512,   3670016,   2097152;
  8: 256,  8192, 114688,  917504,  4587520, 14680064,  29360128,  33554432,  16777216;
  9: 512, 18432, 294912, 2752512, 16515072, 66060288, 176160768, 301989888, 301989888, 134217728;

The row polynomials satisfy P(n,x) = (2+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + 8*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] = (2 + 8*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:   2,     8;
  2:   4,    32,     64;
  3:   8,    96,    384,     512;
  4:  16,   256,   1536,    4096,     4096;
  5:  32,   640,   5120,   20480,    40960,    32768;
  6:  64,  1536,  15360,   81920,   245760,   393216,    262144;
  7: 128,  3584,  43008,  286720,  1146880,  2752512,   3670016,   2097152;
  8: 256,  8192, 114688,  917504,  4587520, 14680064,  29360128,  33554432,  16777216;
  9: 512, 18432, 294912, 2752512, 16515072, 66060288, 176160768, 301989888, 301989888, 134217728;

The row polynomials satisfy P(n,x) = (2+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + 8*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] = (2 + 8*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:         8,         2;
  2:        64,        32,         4;
  3:       512,       384,        96,         8;
  4:      4096,      4096,      1536,       256,       16;
  5:     32768,     40960,     20480,      5120,      640,       32;
  6:    262144,    393216,    245760,     81920,    15360,     1536,      64;
  7:   2097152,   3670016,   2752512,   1146880,   286720,    43008,    3584,    128;
  8:  16777216,  33554432,  29360128,  14680064,  4587520,   917504,  114688,   8192,   256;
  9: 134217728, 301989888, 301989888, 176160768, 66060288, 16515072, 2752512, 294912, 18432, 512;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + 8*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] = (2 + 8*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-d7d2422771947afc
• 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-d7d2422771947afc
• 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-d7d2422771947afc

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A013611 e-d7d2422771947afc

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-7963c2f0e7b86ba8 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-37a63c273567eea5

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-7963c2f0e7b86ba8 (expanded above)

Related sequences ?

Same fingerprint (14)

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

• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• A027467 → basis A013611
• A027466 → basis A013609
• A038231 → D
• basis-map coefficients of A026729, shifted → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• A136572 → basis A013611
• basis-map coefficients of A026729 → basis A013611
• basis-map coefficients of A021012 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013611
• A038330 → basis A007318
• basis-map coefficients of A112227 → basis A013611
• A013611 → D