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n>=0.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:         1;\n  1:         3,          20,           32;\n  2:        26,         300,         1264,          2304,           1536;\n  3:       252,        4128,        27696,         97472,         189952,          194560,            81920;\n  4:      2568,       54480,       499488,       2586048,        8274176,        16762880,         21012480,         14909440,           4587520;\n  5:     26928,      701376,      8143488,      55522304,      246258432,       742712320,       1543192576,       2182053888,        2010251264,        1089994752,          264241152;\n  6:    287648,     8881600,    124736512,    1053918720,     5968017920,     23867730944,      69144309760,     146238898176,      224164577280,      242936709120,       176733290496,        77510737920,        15502147584;\n  7:   3112896,   111150080,   1830950144,   18446121984,   127000011776,    632239382528,    2347403980800,    6604617531392,    14153422733312,    22993104273408,     27880379645952,     24472631377920,     14702679687168,      5412464099328,       921270484992;\n  8:  34013312,  1378752768,  26055394816,  304753390592,  2469639309312,  14705116819456,   66561157505024,  233661014458368,   643012724064256,  1391968241319936,   2362839624843264,   3112476623241216,   3119427398664192,   2299836606971904,   1176462409334784,    373114546421760,    55276229099520;\n  9: 374416128, 16985318400, 362135973888, 4822043394048, 44942276960256, 311378589548544, 1662228653998080, 6991486209490944, 23491160237604864, 63530400071548928, 138637613209747456, 243568896680919040, 341992616897806336, 378603380005666816, 323231741831544832, 205372687416033280, 91457591946772480, 25476199811645440, 3341140958904320;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(16*n^2-48*n+32)*x+(16*n^2-48*n+32)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(56*n^2-140*n+56)*x+(64*n^2-160*n+64)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 20*x + 32*x^2, P(2,x) = 26 + 300*x + 1264*x^2 + 2304*x^3 + 1536*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 20*x + 32*x^2;\nP[2, x_] := 26 + 300*x + 1264*x^2 + 2304*x^3 + 1536*x^4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (16*n^2 - 48*n + 32)*x + (16*n^2 - 48*n + 32)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (56*n^2 - 140*n + 56)*x + (64*n^2 - 160*n + 64)*x^2)*P[n - 1, x])/(n^2 - 2*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 20*x + 32*x^2; v[3] = 26 + 300*x + 1264*x^2 + 2304*x^3 + 1536*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (16*n^2 - 48*n + 32)*x + (16*n^2 - 48*n + 32)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (56*n^2 - 140*n + 56)*x + (64*n^2 - 160*n + 64)*x^2)*v[i - 1])/(n^2 - 2*n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-97a17a1dc7998ff8","steps":["basis A013611"],"edges":["e-fc8980c2fe46c42c"],"text":"A316649 \u2192 basis A013611"},"headline":"A316649 \u2192 basis A013611","chain":null}