A060187 -id:A060187 - cp's OEIS frontend

A146745 Coefficients of Pascal's triangle polynomial minus MacMahon polynomial A060187 with minus the first and last row terms and powers of x divided out: f(n)=3^n - 2n - 1; q(x,n)=2^n(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2]; p(x,n)=((q[x, n] - (x + 1)^n)/x - f[n] - f[n]*x^(n - 2))/x.

224, 1672, 1672, 10528, 23528, 10528, 60636, 259688, 259688, 60636, 331584, 2485232, 4674944, 2485232, 331584, 1756304, 21707888, 69413168, 69413168, 21707888, 1756304, 9116096, 178300784, 906923072, 1527092216, 906923072

Offset: 2

Roger L. Bagula, Nov 01 2008

Row sums starting with n=4 are {224, 3344, 44584, 640648, 10308576, 185754720, 3715772120}. First elements in each row are {224, 1672, 1672, 10528, 60636, 331584, 1756304, 9116096}. Subtracting out the row terms gives the middle elements of the difference.

			Triangle starts
{224},
{1672, 1672},
{10528, 23528, 10528},
{60636, 259688, 259688, 60636},
{331584, 2485232, 4674944, 2485232, 331584},
{1756304, 21707888, 69413168, 69413168, 21707888, 1756304},
{9116096, 178300784, 906923072, 1527092216, 906923072, 178300784, 9116096}

Cf. A061981, A060187.

q[x_, n_] = 2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2]; p[x_, n_] = ((q[x, n] - (x + 1)^n)/x - f[n] - f[n]*x^(n - 2))/x; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 4, 10}]; Flatten[%]

f(n) = 3^n - 2*n - 1;
q(x,n) = 2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2];
p(x,n) = ((q[x, n] - (x + 1)^n)/x - f[n] - f[n]*x^(n - 2))/x;
t(n,m) = Coefficients(p(x,n)) with n starting at 4.

A154817 Triangle T(n,k) = A060187(n+2,k+2), 1<=k<=n.

Original entry on oeis.org

6, 23, 23, 76, 230, 76, 237, 1682, 1682, 237, 722, 10543, 23548, 10543, 722, 2179, 60657, 259723, 259723, 60657, 2179, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 19673, 1756340, 21707972, 69413294, 69413294, 21707972, 1756340

Offset: 1

Roger L. Bagula, Jan 15 2009

The triangle of MacMahon numbers with the first column and diagonal removed.

Row sums are 6, 46, 382, .. = A000165(n+1)-2.

			6;
23, 23;
76, 230, 76;
237, 1682, 1682, 237;
722, 10543, 23548, 10543, 722;
2179, 60657, 259723, 259723, 60657, 2179;
6552, 331612, 2485288, 4675014, 2485288, 331612, 6552;
19673, 1756340, 21707972, 69413294, 69413294, 21707972, 1756340, 19673;
59038, 9116141, 178300904, 906923282, 1527092468, 906923282, 178300904, 9116141, 59038;

Cf. A060187

p[x_, n_] = 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2];
t[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m]];
Table[ Select[ Table[ t[ n, i ], {i, 1, n}], # > 1 & ], {n, 0, 14} ];
Select[ Flatten[ Table[ t[ n, i ], {n, 0, 13}, {i, 1, n} ] ], # > 1 & ]

A155558 Multi-bifurcating recursion of a factorial type based on the MacMahon numbers A060187 as a triangle sequence: t(n,k) = A060187(n,m) from polynomials; f(n, m) = If[m <= Floor[n/2], f(m, 1)f(n - m, 1)t(n + 1, m)].

Original entry on oeis.org

1, 1, 6, 138, 10488, 8280, 2485656, 1392696, 1794643632, 663449904, 448448112, 3910528474128, 904634615952, 375908525712, 25621782562486656, 3570752184568704, 852504801992064, 514242863174016, 504057328351799983488

Offset: 0

Roger L. Bagula, Jan 24 2009

Row sums are: {1, 1, 6, 138, 18768, 3878352, 2906541648, 5191071615792, 30559282412221440, 552452422047369288192, 31282693282240728431406336,...}.
The Eulerian numbers factored as factorial like to middle Floor[n/2]: t(n,m)=f(n,m)/(f[m,1]*f[n-m,1]).
The idea is to factor the Eulerian numbers as if the coefficients were made up of equivalents to factorials.
The result is a multi-bifurcating recursion thast fits the Eulerian numbers.

			Half MacMahon numbers: Table[Table[f[n, m]/(f[m, 1]*f[n - m, 1]), {m, 0, Floor[n/2]}], {n, 0, 10}];
  {1},
  {1},
  {1, 6},
  {1, 23},
  {1, 76, 230},
  {1, 237, 1682},
  {1, 722, 10543, 23548},
  {1, 2179, 60657, 259723},
  {1, 6552, 331612, 2485288, 4675014},
  {1, 19673, 1756340, 21707972, 69413294},
  {1, 59038, 9116141, 178300904, 906923282, 1527092468}
Factorial type triangle is:
  {1},
  {1},
  {6},
  {138},
  {10488, 8280},
  {2485656, 1392696},
  {1794643632, 663449904, 448448112},
  {3910528474128, 904634615952, 375908525712},
  {25621782562486656, 3570752184568704, 852504801992064, 514242863174016},
  ...

Cf. A008292, A060187.

Clear[t, n, m, f, x, p];
p[x_, n_] = (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2* m + 1)^n*x^m, {m, 0, Infinity}];
t[n_, m_] := Table[CoefficientList[FullSimplify[ExpandAll[p[x, k]]], x], { k, 0, 10}][[n + 1, m + 1]];
f[0, 1] = 1; f[1, 1] = 1; f[2, 1] = 6;
f[n_, m_] := f[n, m] = If[m <= Floor[n/2], f[m, 1]*f[n - m, 1]*t[n, m]];
a = Join[{{1}}, {{1}}, Table[Table[f[n, m], {m, 1, Floor[n/2]}], {n, 2, 10}]];
Flatten[%]

t(n,k)=A060187[n,m] from polynomials; f(n, m) = If[m <= Floor[n/2], f(m, 1)*f(n - m, 1)*t(n + 1, m)];

A155871 Subtraction of polynomial coefficients of MacMahon A060187 from third derivative of Pascal's triangle A155863: p(x,n)=(If[n == 0, 1, x^n + 1 + xD[( x + 1)^(n + 1), {x, 3}]] - 2^n(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x.

Original entry on oeis.org

1, 1, -16, -110, -16, -117, -1322, -1322, -117, -512, -9703, -22288, -9703, -512, -1843, -58977, -256363, -256363, -58977, -1843, -6048, -328588, -2477728, -4664934, -2477728, -328588, -6048, -18953, -1751300, -21692852, -69388094

Offset: 3

Roger L. Bagula, Jan 29 2009

Row sums are:
{2, -142, -2878, -42718, -634366, -10289662, -185702398, -3715637758,
-81748930558, -1961988796414, -51011749920766}

			{1, 1},
{-16, -110, -16},
{-117, -1322, -1322, -117},
{-512, -9703, -22288, -9703, -512},
{-1843, -58977, -256363, -256363, -58977, -1843},
{-6048, -328588, -2477728, -4664934, -2477728, -328588, -6048},
{-18953, -1751300, -21692852, -69388094, -69388094, -21692852, -1751300, -18953},
{-58048, -9108221, -178273184, -906867842, -1527023168, -906867842, -178273184, -9108221, -58048},
{-175815, -46690547, -1403033205, -10836712218, -28587853494, -28587853494, -10836712218, -1403033205, -46690547, -175815},
{-529712, -237214810, -10708833968, -121383574287, -477020204064, -743288082732, -477020204064, -121383574287, -10708833968, -237214810, -529712},
{-1592125, -1198358670, -79944129566, -1295922974075, -7310749751463, -16818058154484, -16818058154484, -7310749751463, -1295922974075, -79944129566, -1198358670, -1592125}

A060187, A155863

p[x_, n_] = (If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n + 1), {x, 3}]] - 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x;
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 3, 13}];
a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 3, 13}];
Flatten[a]

p(x,n)=(If[n == 0, 1, x^n + 1 + x*D[( x + 1)^(n + 1), {x, 3}]]
- 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x;
t(n,m)=coefficients(p(x,n))

A158781 Triangle: A060187 with interspersed zeros.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 6, 0, 1, 1, 0, 23, 0, 23, 0, 1, 1, 0, 76, 0, 230, 0, 76, 0, 1, 1, 0, 237, 0, 1682, 0, 1682, 0, 237, 0, 1, 1, 0, 722, 0, 10543, 0, 23548, 0, 10543, 0, 722, 0, 1, 1, 0, 2179, 0, 60657, 0, 259723, 0, 259723, 0, 60657, 0, 2179, 0, 1, 1, 0, 6552, 0, 331612, 0, 2485288

Offset: 0

Roger L. Bagula, Mar 26 2009

			{1},
{1, 0, 1},
{1, 0, 6, 0, 1},
{1, 0, 23, 0, 23, 0, 1},
{1, 0, 76, 0, 230, 0, 76, 0, 1},
{1, 0, 237, 0, 1682, 0, 1682, 0, 237, 0, 1},
{1, 0, 722, 0, 10543, 0, 23548, 0, 10543, 0, 722, 0, 1},
...

Cf. A060187.

A171273 Matrix inverse of A060187.

Original entry on oeis.org

1, 1, 1, 5, 6, 1, 93, 115, 23, 1, 5993, 7436, 1518, 76, 1, 1272089, 1578757, 322762, 16330, 237, 1, 857402029, 1064110290, 217560951, 11012540, 160571, 722, 1, 1792650585525, 2224835452407, 454875884137, 23025275075, 335768223, 1512581, 2179, 1

Offset: 1

Roger L. Bagula and Mats Granvik, Dec 06 2009

			{1},
{1, 1},
{5, 6, 1},
{93, 115, 23, 1},
{5993, 7436, 1518, 76, 1},
{1272089, 1578757, 322762, 16330, 237, 1},
{857402029, 1064110290, 217560951, 11012540, 160571, 722, 1},
{1792650585525, 2224835452407, 454875884137, 23025275075, 335768223, 1512581, 2179, 1},
{11464255554367057, 14228139328931096, 2908996087466828, 147249943814184, 2147290464886, 9673492136, 13945196, 6552, 1},
{222406320165016449457, 276025608122908733321, 56434463826320585284, 2856645864675796564, 41657391444153086, 187665608020478, 270538484020, 127141156, 19673, 1},
{13026233415367869864109781, 16166689855580307839632286, 3305339964838291288943901, 167312402773377971746920, 2439853795947184617546, 10991486289326969076, 15845312257310658, 7446608913000, 1152338433, 59038, 1}

Cf. A060187, A055325, A154921,

m = 2;
A[n_, 1] := 1
A[n_, n_] := 1
A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
a = Table[A[n, k], {n, 12}, {k, n}]
M[n_] := Table[If[k <= m, (-1)^(m + k)*a[[m, k]], 0], {k, 1, n}, {m, 1, n}]
Table[Table[Inverse[M[12]][[m, n]], {m, 1, n}], {n, 1, 11}]
Flatten[%]

A(n,k) = (m*n - m*k + 1) * A(n - 1, k - 1) + (m*k - (m - 1)) * A(n - 1, k), A(n,1) = A(n,n) = 1.

A176467 A symmetrical triangle:q=2;c(n,q)=Product[1 - q^i, {i, 1, n}];t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 17, 17, 1, 1, 62, 196, 62, 1, 1, 207, 1528, 1528, 207, 1, 1, 660, 9893, 22154, 9893, 660, 1, 1, 2053, 57991, 247913, 247913, 57991, 2053, 1, 1, 6298, 320818, 2388134, 4474228, 2388134, 320818, 6298, 1, 1, 19163, 1712906, 20919938

Offset: 0

Roger L. Bagula, Apr 18 2010

Row sums are:

{1, 2, 6, 36, 322, 3472, 43262, 615916, 9904730, 177511112, 3486135606,...}.

			{1},
{1, 1},
{1, 4, 1},
{1, 17, 17, 1},
{1, 62, 196, 62, 1},
{1, 207, 1528, 1528, 207, 1},
{1, 660, 9893, 22154, 9893, 660, 1},
{1, 2053, 57991, 247913, 247913, 57991, 2053, 1},
{1, 6298, 320818, 2388134, 4474228, 2388134, 320818, 6298, 1},
{1, 19163, 1712906, 20919938, 66103548, 66103548, 20919938, 1712906, 19163, 1},
{1, 58016, 8941891, 171953190, 853179296, 1417870818, 853179296, 171953190, 8941891, 58016, 1}

Cf. A060187

(*A060187*);
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
c[n_, q_] = Product[1 - q^i, {i, 1, n}];
t[n_, m_, q_] := f[n, m] - c[n, q]/(c[m, q]*c[n - m, q]) + 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]

q=2;
c(n,q)=Product[1 - q^i, {i, 1, n}];
t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1

A176468 A symmetrical triangle:q=3;c(n,q)=Product[1 - q^i, {i, 1, n}];t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 11, 11, 1, 1, 37, 101, 37, 1, 1, 117, 473, 473, 117, 1, 1, 359, -467, -10331, -467, 359, 1, 1, 1087, -38805, -666047, -666047, -38805, 1087, 1, 1, 3273, -564647, -22609991, -71238207, -22609991, -564647, 3273, 1, 1, 9833, -6313279

Offset: 0

Roger L. Bagula, Apr 18 2010

Row sums are:
{1, 2, 5, 24, 177, 1182, -10545, -1407528, -117580935, -13535434518,
-2541110736213,...}.

			{1},
{1, 1},
{1, 3, 1},
{1, 11, 11, 1},
{1, 37, 101, 37, 1},
{1, 117, 473, 473, 117, 1},
{1, 359, -467, -10331, -467, 359, 1},
{1, 1087, -38805, -666047, -666047, -38805, 1087, 1},
{1, 3273, -564647, -22609991, -71238207, -22609991, -564647, 3273, 1},
{1, 9833, -6313279, -656760847, -6104652967, -6104652967, -656760847, -6313279, 9833, 1},
{1, 29515, -63520279, -18148426855, -499870912759, -1504945075459, -499870912759, -18148426855, -63520279, 29515, 1}

Cf. A060187

(*A060187*);
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
c[n_, q_] = Product[1 - q^i, {i, 1, n}];
t[n_, m_, q_] := f[n, m] - c[n, q]/(c[m, q]*c[n - m, q]) + 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]

q=3;
c(n,q)=Product[1 - q^i, {i, 1, n}];
t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1

A176469 A symmetrical triangle:q=4;c(n,q)=Product[1 - q^i, {i, 1, n}];t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, -8, -126, -8, 1, 1, -103, -4114, -4114, -103, 1, 1, -642, -82549, -353256, -82549, -642, 1, 1, -3281, -1430195, -23948889, -23948889, -1430195, -3281, 1, 1, -15292, -23527496, -1548356796, -6216938526

Offset: 0

Roger L. Bagula, Apr 18 2010

Row sums are:
{1, 2, 4, 8, -140, -8432, -519636, -50764728, -9360737692, -3387701237632,
-246332974040099,...}.

			{1},
{1, 1},
{1, 2, 1},
{1, 3, 3, 1},
{1, -8, -126, -8, 1},
{1, -103, -4114, -4114, -103, 1},
{1, -642, -82549, -353256, -82549, -642, 1},
{ 1, -3281, -1430195, -23948889, -23948889, -1430195, -3281, 1},
{1, -15292, -23527496, -1548356796, -6216938526, -1548356796, -23527496, -15292, 1},
{1, -67707, -380011248, -99256044576, -1594214495286, -1594214495286, -99256044576, -380011248, -67707, 1},
{1, -290486, -6099252663, -6353979629820, -408235051426002, -1634139479203056, -408235051426002, -6353979629820, -6099252663, -290486, 1}

Cf. A060187

(*A060187*);
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
c[n_, q_] = Product[1 - q^i, {i, 1, n}];
t[n_, m_, q_] := f[n, m] - c[n, q]/(c[m, q]*c[n - m, q]) + 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]

q=4;
c(n,q)=Product[1 - q^i, {i, 1, n}];
t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1

A177826 Sub-triangle of A060187: even-indexed entries of even-indexed rows.

Original entry on oeis.org

1, 1, 1, 1, 230, 1, 1, 10543, 10543, 1, 1, 331612, 4675014, 331612, 1, 1, 9116141, 906923282, 906923282, 9116141, 1, 1, 237231970, 121383780207, 743288515164, 121383780207, 237231970, 1, 1, 6031771195, 13342139253321, 342917527152507, 342917527152507, 13342139253321, 6031771195, 1

Offset: 0

Roger L. Bagula, Dec 13 2010

			Triangle begins:
  {1},
  {1, 1},
  {1, 230, 1},
  {1, 10543, 10543, 1},
  {1, 331612, 4675014, 331612, 1},
  {1, 9116141, 906923282, 906923282, 9116141, 1},

Cf. A060187

p[x_, n_] = (1 - x)^(n + 1)*Sum[((2*k + 1)^n)*x^k, {k, 0, Infinity}];
t[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
Table[Table[t[n, 2*m], {m, 0, Floor[n/2]}], {n, 0, 20, 2}];
Flatten[%]
(* Alternative recursion for A060187 *)
m = 2;
A[n_, 1] := 1
A[n_, n_] := 1
A[n_, k_] := A[n, k] = (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
Table[A[n,k],{n,1,10,2},{k,1,n,2}]
(* Alternative expansion for A060187 *)
p[t_] = Exp[t] *x/(-Exp[2*t] + x)
Table[ CoefficientList[FullSimplify[ExpandAll[(n!*(-1 + x)^(n + \
1)/x)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]

cp's OEIS Frontend

A146745 Coefficients of Pascal's triangle polynomial minus MacMahon polynomial A060187 with minus the first and last row terms and powers of x divided out: f(n)=3^n - 2*n - 1; q(x,n)=2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2]; p(x,n)=((q[x, n] - (x + 1)^n)/x - f[n] - f[n]*x^(n - 2))/x.

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A154817 Triangle T(n,k) = A060187(n+2,k+2), 1<=k<=n.

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A155558 Multi-bifurcating recursion of a factorial type based on the MacMahon numbers A060187 as a triangle sequence: t(n,k) = A060187(n,m) from polynomials; f(n, m) = If[m <= Floor[n/2], f(m, 1)*f(n - m, 1)*t(n + 1, m)].

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A155871 Subtraction of polynomial coefficients of MacMahon A060187 from third derivative of Pascal's triangle A155863: p(x,n)=(If[n == 0, 1, x^n + 1 + x*D[( x + 1)^(n + 1), {x, 3}]] - 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x.

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A158781 Triangle: A060187 with interspersed zeros.

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A171273 Matrix inverse of A060187.

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A176467 A symmetrical triangle:q=2;c(n,q)=Product[1 - q^i, {i, 1, n}];t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1.

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A176468 A symmetrical triangle:q=3;c(n,q)=Product[1 - q^i, {i, 1, n}];t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1.

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A176469 A symmetrical triangle:q=4;c(n,q)=Product[1 - q^i, {i, 1, n}];t(n,m,q)=A060187(n,m)-c(n,q)/(c(m,q)*c(n-m,q))+1.

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A146745 Coefficients of Pascal's triangle polynomial minus MacMahon polynomial A060187 with minus the first and last row terms and powers of x divided out: f(n)=3^n - 2n - 1; q(x,n)=2^n(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2]; p(x,n)=((q[x, n] - (x + 1)^n)/x - f[n] - f[n]*x^(n - 2))/x.

A155558 Multi-bifurcating recursion of a factorial type based on the MacMahon numbers A060187 as a triangle sequence: t(n,k) = A060187(n,m) from polynomials; f(n, m) = If[m <= Floor[n/2], f(m, 1)f(n - m, 1)t(n + 1, m)].

A155871 Subtraction of polynomial coefficients of MacMahon A060187 from third derivative of Pascal's triangle A155863: p(x,n)=(If[n == 0, 1, x^n + 1 + xD[( x + 1)^(n + 1), {x, 3}]] - 2^n(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x.