A109626 -id:A109626 - cp's OEIS frontend

A111618 First lower diagonal of A109626.

1, 3, 2, 5, 3, 7, 4, 9, 10, 11, 6, 13, 14, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12, 25, 26, 27, 14, 29, 30, 31, 16, 33, 17, 35, 18, 37, 38, 39, 20, 41, 42, 43, 22, 45, 46, 47, 48, 49, 50, 51, 52, 53, 27, 55, 56, 57, 29, 59, 30, 61, 31, 63, 32, 65, 66, 67, 34, 69, 70, 71, 72, 73, 74

Offset: 2

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

nonn

The odd-indexed bisection is a(2n+1)=2n+1.

The even-indexed bisection: A111619.

Cf. A109626, A111619, A111620.

f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 80}]]; g[n_, m_] := f[n][[m]]; Table[ g[n, n - 1], {n, 2, 74}]

A111615 Third upper diagonal of the array in A109626.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 12, 13, 14, 15, 16, 17, 9, 19, 10, 21, 11, 23, 24, 25, 26, 27, 28, 29, 15, 31, 32, 33, 17, 35, 18, 37, 19, 39, 40, 41, 42, 43, 22, 45, 46, 47, 48, 49, 25, 51, 26, 53, 54, 55, 28, 57, 29, 59, 60, 61, 31, 63, 64, 65, 66, 67, 68, 69, 70, 71, 36, 73

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Jul 30 2005

nonn

Cf. A109626.

f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 80}]]; g[n_, m_] := f[n][[m]]; Table[ g[n, n + 3], {n, 74}]

A111607 Fourth column of A109626.

Original entry on oeis.org

1, 2, 3, 3, 5, 3, 7, 2, 9, 10, 11, 9, 13, 7, 15, 4, 17, 18, 19, 15, 21, 11, 23, 6, 25, 26, 27, 21, 29, 15, 31, 8, 33, 34, 35, 27, 37, 19, 39, 10, 41, 42, 43, 33, 45, 23, 47, 12, 49, 50, 51, 39, 53, 27, 55, 14, 57, 58, 59, 45, 61, 31, 63, 16, 65, 66, 67, 51, 69, 35, 71, 18, 73, 74, 75

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A109626.

R:=PowerSeriesRing(Integers(), 102);
p:= func< x | x*(1+2*x+3*x^2+3*x^3 +5*x^4 +3*x^5 +7*x^6 +2*x^7 +7*x^8 +6*x^9 +5*x^10 +3*x^11 +3*x^12 +x^13 +x^14)/(1-x^8)^2 >;
Coefficients(R!( p(x) )); // G. C. Greubel, Jan 29 2025

(* First program *)
f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 128}]]; g[n_, m_] := f[n][[m]]; Table[g[n, 4 + 1], {n, 75}]
(* Second program *)
CoefficientList[Series[(1+2*x+3*x^2+3*x^3+5*x^4+3*x^5+7*x^6+2*x^7+7*x^8 +6*x^9+5*x^10+3*x^11+3*x^12+x^13+x^14)/(1-x^8)^2, {x,0,100}], x] (* G. C. Greubel, Jan 29 2025 *)

def p(x): return x*(1+2*x+3*x^2+3*x^3 +5*x^4 +3*x^5 +7*x^6 +2*x^7 +7*x^8 +6*x^9 +5*x^10 +3*x^11 +3*x^12 +x^13 +x^14)
def A111607_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( p(x)/(1-x^8)^2 ).list()
a=A111607_list(101); a[1:] # G. C. Greubel, Jan 29 2025

G.f.: x*(1 + 2*x + 3*x^2 + 3*x^3 + 5*x^4 + 3*x^5 + 7*x^6 + 2*x^7 + 7*x^8 + 6*x^9 + 5*x^10 + 3*x^11 + 3*x^12 + x^13 + x^14)/(1-x^8)^2.

A111627 T(2n, n+2)/n of A109626.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 02 2005

nonn

A sequence of just 1's and 2's.

Cf. A109626.

f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 128}]]; g[n_, m_] := f[n][[m]]; Table[ g[2n, n + 2]/n, {n, 52}]

A111605 Determinant of the upper left n X n elements of the array T(n, m) in A109626.

Original entry on oeis.org

1, 1, 2, 2, 8, -40, 144, 2448, -17280, -71424, 0, 1198080, 0, -18063360, 418037760, -7498137600, -115688079360, -587977850880, 12300343050240, -540240522117120, -36548032147292160, 487948583538524160, -2339225182084792320

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

sign

a(n)=0 for n's: 11, 13, 29, 31, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 48, 49, 50, 51, 53,59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 80

Cf. A109626, A111606.

f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 32}]]; g[n_, m_] := f[n][[m]]; Table[ Det[ Table[ g[i, j], {i, n}, {j, n}]], {n, 23}]

A111606 Determinant of the upper left n X n elements excluding its first row and column of the Array T(n,m) in A109626.

Original entry on oeis.org

2, 3, 0, 10, -48, 168, 3456, -29376, -178560, 0, 2995200, 0, -45158400, 1045094400, -14636482560, -122918584320, -370845941760, 12983695441920, -1545691830681600, -62521904633610240, 785989658251100160, -2445553599452282880

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

sign

a(n)=0 for n's, (one less than those in A111605): 3, 10, 12, 28, 30, 33, 34, 35, 36, 37, 38, 40, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 79

Cf. A109626, A111605.

f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 32}]]; g[n_, m_] := f[n][[m]]; Table[ Det[ Table[ g[i, j], {i, 2, n}, {j, 2, n}]]{n, 23}]

A111608 Sixth column of A109626.

Original entry on oeis.org

1, 1, 3, 4, 5, 6, 7, 4, 3, 5, 11, 4, 13, 14, 10, 8, 17, 3, 19, 20, 21, 22, 23, 12, 25, 13, 9, 28, 29, 10, 31, 16, 22, 17, 35, 24, 37, 38, 39, 20, 41, 21, 43, 44, 15, 46, 47, 40, 49, 25, 34, 52, 53, 36, 55, 28, 57, 29, 59, 60, 61, 62, 21, 32, 65, 55, 67, 68, 46, 70, 71, 12, 73, 37

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

nonn

Cf. A109626.

f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 128}]]; g[n_, m_] := f[n][[m]]; Table[g[n, 6 + 1], {n, 74}]

G.f.: Q(x)/(1-x^72)^2 where Q(x) is 142-degree polynomial: Q(x)=1+x+3x^2+4x^3+5x^4+6x^5+7x^6+4x^7+...

A111609 Eighth column of A109626.

Original entry on oeis.org

1, 2, 3, 1, 5, 3, 7, 7, 9, 10, 11, 12, 13, 7, 15, 10, 17, 18, 19, 15, 21, 11, 23, 9, 25, 26, 27, 14, 29, 15, 31, 4, 33, 34, 35, 9, 37, 19, 39, 35, 41, 42, 43, 44, 45, 23, 47, 30, 49, 50, 51, 39, 53, 27, 55, 21, 57, 58, 59, 30, 61, 31, 63, 8, 65, 66, 67, 17, 69, 35, 71, 63, 73, 74, 75

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

nonn

Cf. A109626.

f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 128}]]; g[n_, m_] := f[n][[m]]; Table[ g[n, 8 + 1], {n, 111}]

A111610 Ninth column of A109626.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 1, 10, 11, 8, 13, 14, 10, 16, 17, 2, 19, 20, 14, 22, 23, 24, 25, 26, 3, 28, 29, 10, 31, 32, 33, 34, 35, 4, 37, 38, 13, 40, 41, 14, 43, 44, 5, 46, 47, 48, 49, 50, 17, 52, 53, 6, 55, 56, 57, 58, 59, 40, 61, 62, 7, 64, 65, 44, 67, 68, 46, 70, 71, 8, 73, 74, 50

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

nonn

Cf. A109626.

f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 128}]]; g[n_, m_] := f[n][[m]]; Table[ g[n, 9 + 1], {n, 75}]

A111611 Tenth column of A109626.

Original entry on oeis.org

1, 1, 3, 4, 4, 6, 7, 8, 9, 3, 11, 6, 13, 14, 15, 8, 17, 9, 19, 20, 21, 22, 23, 24, 5, 13, 27, 14, 29, 6, 31, 16, 33, 17, 14, 36, 37, 38, 39, 16, 41, 21, 43, 22, 27, 46, 47, 24, 49, 5, 51, 52, 53, 54, 44, 56, 57, 29, 59, 18, 61, 62, 63, 32, 65, 33, 67, 68, 69, 70, 71, 72, 73, 37, 15

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

nonn

Cf. A109626.

f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 128}]]; g[n_, m_] := f[n][[m]]; Table[g[n, 10 + 1], {n, 75}]

cp's OEIS Frontend

A111618 First lower diagonal of A109626.

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A111615 Third upper diagonal of the array in A109626.

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A111607 Fourth column of A109626.

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A111627 T(2n, n+2)/n of A109626.

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A111605 Determinant of the upper left n X n elements of the array T(n, m) in A109626.

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A111606 Determinant of the upper left n X n elements excluding its first row and column of the Array T(n,m) in A109626.

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A111608 Sixth column of A109626.

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A111609 Eighth column of A109626.

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A111610 Ninth column of A109626.

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A111611 Tenth column of A109626.

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