A013616 -id:A013616 - cp's OEIS frontend

A038291 Triangle whose (i,j)-th entry is binomial(i,j)9^(i-j)1^j.

1, 9, 1, 81, 18, 1, 729, 243, 27, 1, 6561, 2916, 486, 36, 1, 59049, 32805, 7290, 810, 45, 1, 531441, 354294, 98415, 14580, 1215, 54, 1, 4782969, 3720087, 1240029, 229635, 25515, 1701, 63, 1, 43046721, 38263752, 14880348, 3306744, 459270, 40824, 2268, 72, 1

Offset: 0

N. J. A. Sloane

T(i,j) is the number of i-permutations of 10 objects a,b,c,d,e,f,g,h,i,j with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007
Reflected version of A013616. - R. J. Mathar, Dec 19 2008
Triangle of coefficients in expansion of (9 + x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 21 2018

			Triangle begins:
  1
  9, 1
  81, 18, 1
  729, 243, 27, 1
  6561, 2916, 486, 36, 1
  59049, 32805, 7290, 810, 45, 1
  531441, 354294, 98415, 14580, 1215, 54, 1
  4782969, 3720087, 1240029, 229635, 25515, 1701, 63, 1
  43046721, 38263752, 14880348, 3306744, 459270, 40824, 2268, 72, 1

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Muniru A Asiru, Rows n=0..50 of triangle, flattened
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A317051, A317052.

Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*9^(i-j)*1^j))); # Muniru A Asiru, Jul 21 2018

for i from 0 to 9 do seq(binomial(i, j)*9^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 9 t[n - 1, k] + t[n - 1, k - 1]];
Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Zagros Lalo, Jul 21 2018 *)
Table[CoefficientList[ Expand[(9 + x)^n], x], {n, 0, 8}] // Flatten  (* Zagros Lalo, Jul 22 2018 *)

T(0,0) = 1; T(n,k) = 9*T(n-1,k) + T(n-1,k-1) for k = 0..n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 21 2018

A317051 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 9 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, 9, 1, 18, 1, 27, 81, 1, 36, 243, 1, 45, 486, 729, 1, 54, 810, 2916, 1, 63, 1215, 7290, 6561, 1, 72, 1701, 14580, 32805, 1, 81, 2268, 25515, 98415, 59049, 1, 90, 2916, 40824, 229635, 354294, 1, 99, 3645, 61236, 459270, 1240029, 531441, 1, 108, 4455, 87480, 826686, 3306744, 3720087

Offset: 0

Zagros Lalo, Jul 20 2018

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013616 ((1+9*x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A038291 ((9+x)^n).
The coefficients in the expansion of 1/(1-x-9*x^2) are given by the sequence generated by the row sums.
The row sums are Generalized Fibonacci numbers (see A015445).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.5413812651491... ((1+sqrt(37))/2), when n approaches infinity.

			Triangle begins:
  1;
  1;
  1, 9;
  1, 18;
  1, 27, 81;
  1, 36, 243;
  1, 45, 486, 729;
  1, 54, 810, 2916;
  1, 63, 1215, 7290, 6561;
  1, 72, 1701, 14580, 32805;
  1, 81, 2268, 25515, 98415, 59049;
  1, 90, 2916, 40824, 229635, 354294;
  1, 99, 3645, 61236, 459270, 1240029, 531441;
  1, 108, 4455, 87480, 826686, 3306744, 3720087;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 100

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 9x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (9 + x)^n

Row sums give A015445.

Cf. A013616, A038291.

Flat(List([0..13],n->List([0..Int(n/2)],k->9^k*Binomial(n-k,k)))); # Muniru A Asiru, Jul 20 2018

/* As triangle */ [[9^k*Binomial(n-k,k): k in [0..Floor(n/2)]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 05 2018

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 9 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten
Table[9^k Binomial[n - k, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = 9^k*binomial(n-k,k);
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

T(n,k) = 9^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).

A317052 Triangle read by rows: T(0,0) = 1; T(n,k) = 9*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 9, 81, 1, 729, 18, 6561, 243, 1, 59049, 2916, 27, 531441, 32805, 486, 1, 4782969, 354294, 7290, 36, 43046721, 3720087, 98415, 810, 1, 387420489, 38263752, 1240029, 14580, 45, 3486784401, 387420489, 14880348, 229635, 1215, 1, 31381059609, 3874204890, 172186884, 3306744, 25515, 54

Offset: 0

Zagros Lalo, Jul 20 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013616 ((1+9*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A038291 ((9+x)^n).
The coefficients in the expansion of 1/(1-9*x-x^2) are given by the sequence generated by the row sums (see A099371).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 9.109772228646443655... (a metallic mean), when n approaches infinity; (see A176522: ((9+sqrt(85))/2)).

			Triangle begins:
  1;
  9;
  81, 1;
  729, 18;
  6561, 243, 1;
  59049, 2916, 27;
  531441, 32805, 486, 1;
  4782969, 354294, 7290, 36;
  43046721, 3720087, 98415, 810, 1;
  387420489, 38263752, 1240029, 14580, 45;
  3486784401, 387420489, 14880348, 229635, 1215, 1;
  31381059609, 3874204890, 172186884, 3306744, 25515, 54;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 100.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 9x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (9 + x)^n

Row sums give A099371.
Cf. A013616, A038291, A176522.
Cf. A001019 (column 0), A053540 (column 1), A081139 (column 2), A173187 (column 3), A173000 (column 4).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 9 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 9*T(n-1, k)+T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

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A038291 Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.

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A317051 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 9 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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A317052 Triangle read by rows: T(0,0) = 1; T(n,k) = 9*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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A038291 Triangle whose (i,j)-th entry is binomial(i,j)9^(i-j)1^j.