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A303650 G.f.: A(x,y) = (1-y)^2 * Sum_{n>=0} (2*n+1) * y^n * (1 + x*(1-y)^2 )^(n*(n+1)/2). 3
1, 1, 0, 3, 3, 0, 0, 0, 15, 15, 0, 0, 0, 0, 5, 100, 100, 5, 0, 0, 0, 0, 0, 105, 840, 840, 105, 0, 0, 0, 0, 0, 0, 42, 1764, 8589, 8589, 1764, 42, 0, 0, 0, 0, 0, 0, 7, 1792, 29232, 104104, 104104, 29232, 1792, 7, 0, 0, 0, 0, 0, 0, 0, 1080, 53505, 508680, 1463760, 1463760, 508680, 53505, 1080, 0, 0, 0, 0, 0, 0, 0, 0, 405, 63495, 1433205, 9504540, 23457780, 23457780, 9504540, 1433205, 63495, 405, 0, 0, 0, 0, 0, 0, 0, 0, 90, 53255, 2737090, 37539550, 192046210, 422352880, 422352880, 192046210, 37539550, 2737090, 53255, 90, 0, 0, 0, 0, 0, 0, 0, 0, 9, 32835, 3860661, 103586637, 998951415, 4199275509, 8443603509, 8443603509, 4199275509, 998951415, 103586637, 3860661, 32835, 9, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Compare to: Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)/2) = Product_{n>=1} (1 - x^n)^3.

G.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..2*n+1} T(n,k) * x^n*y^k.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..2651 of rows 0..50 as a flattened triangle read by rows.

FORMULA

Row sums = 2*(2*n+1)!/(2^n*n!) = 2*A001147(n+1) for n >= 0, where A001147 is the odd double factorials.

EXAMPLE

G.f.: A(x,y) = (1 + y) + (3*y + 3*y^2)*x + (15*y^2 + 15*y^3)*x^2 + (5*y^2 + 100*y^3 + 100*y^4 + 5*y^5)*x^3 + (105*y^3 + 840*y^4 + 840*y^5 + 105*y^6)*x^4 + (42*y^3 + 1764*y^4 + 8589*y^5 + 8589*y^6 + 1764*y^7 + 42*y^8)*x^5 + (7*y^3 + 1792*y^4 + 29232*y^5 + 104104*y^6 + 104104*y^7 + 29232*y^8 + 1792*y^9 + 7*y^10)*x^6 + (1080*y^4 + 53505*y^5 + 508680*y^6 + 1463760*y^7 + 1463760*y^8 + 508680*y^9 + 53505*y^10 + 1080*y^11)*x^7 + (405*y^4 + 63495*y^5 + 1433205*y^6 + 9504540*y^7 + 23457780*y^8 + 23457780*y^9 + 9504540*y^10 + 1433205*y^11 + 63495*y^12 + 405*y^13)*x^8 + (90*y^4 + 53255*y^5 + 2737090*y^6 + 37539550*y^7 + 192046210*y^8 + 422352880*y^9 + 422352880*y^10 + 192046210*y^11 + 37539550*y^12 + 2737090*y^13 + 53255*y^14 + 90*y^15)*x^9 + (9*y^4 + 32835*y^5 + 3860661*y^6 + 103586637*y^7 + 998951415*y^8 + 4199275509*y^9 + 8443603509*y^10 + 8443603509*y^11 + 4199275509*y^12 + 998951415*y^13 + 103586637*y^14 + 3860661*y^15 + 32835*y^16 + 9*y^17)*x^10 + ...

This table begins:

[1, 1];

[0, 3, 3, 0];

[0, 0, 15, 15, 0, 0];

[0, 0, 5, 100, 100, 5, 0, 0];

[0, 0, 0, 105, 840, 840, 105, 0, 0, 0];

[0, 0, 0, 42, 1764, 8589, 8589, 1764, 42, 0, 0, 0];

[0, 0, 0, 7, 1792, 29232, 104104, 104104, 29232, 1792, 7, 0, 0, 0];

[0, 0, 0, 0, 1080, 53505, 508680, 1463760, 1463760, 508680, 53505, 1080, 0, 0, 0, 0];

[0, 0, 0, 0, 405, 63495, 1433205, 9504540, 23457780, 23457780, 9504540, 1433205, 63495, 405, 0, 0, 0, 0];

[0, 0, 0, 0, 90, 53255, 2737090, 37539550, 192046210, 422352880, 422352880, 192046210, 37539550, 2737090, 53255, 90, 0, 0, 0, 0];

[0, 0, 0, 0, 9, 32835, 3860661, 103586637, 998951415, 4199275509, 8443603509, 8443603509, 4199275509, 998951415, 103586637, 3860661, 32835, 9, 0, 0, 0, 0];

[0, 0, 0, 0, 0, 15015, 4224948, 216209448, 3718139880, 27514752720, 99160777170, 185620024044, 185620024044, 99160777170, 27514752720, 3718139880, 216209448, 4224948, 15015, 0, 0, 0, 0, 0];

[0, 0, 0, 0, 0, 5005, 3690960, 358604610, 10642877700, 131644916130, 791740742520, 2520648738335, 4450814005365, 4450814005365, 2520648738335, 791740742520, 131644916130, 10642877700, 358604610, 3690960, 5005, 0, 0, 0, 0, 0];

[0, 0, 0, 0, 0, 1155, 2613030, 487998630, 24553882080, 490836585960, 4698816262560, 23906703844170, 68728774397430, 115607871091860, 115607871091860, 68728774397430, 23906703844170, 4698816262560, 490836585960, 24553882080, 487998630, 2613030, 1155, 0, 0, 0, 0, 0];

[0, 0, 0, 0, 0, 165, 1506690, 556471575, 47141475900, 1490071870665, 21979787337360, 171224451938295, 758839035482175, 2002890837642615, 3233811469903935, 3233811469903935, 2002890837642615, 758839035482175, 171224451938295, 21979787337360, 1490071870665, 47141475900, 556471575, 1506690, 165, 0, 0, 0, 0, 0];

[0, 0, 0, 0, 0, 11, 705080, 539064032, 77031946432, 3798613216696, 84402539088022, 976447523140464, 6417840035895872, 25328044049941568, 62168295500414400, 96919878129098048, 96919878129098048, 62168295500414400, 25328044049941568, 6417840035895872, 976447523140464, 84402539088022, 3798613216696, 77031946432, 539064032, 705080, 11, 0, 0, 0, 0, 0]; ...

CROSSREFS

Cf. A303651, A303652.

Sequence in context: A144331 A216805 A167259 * A248625 A000876 A287712

Adjacent sequences:  A303647 A303648 A303649 * A303651 A303652 A303653

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 30 2018

STATUS

approved

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Last modified November 20 21:03 EST 2019. Contains 329348 sequences. (Running on oeis4.)