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binomial(i,j)*3^(i-j)*8^j.","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:           825,            2752,             1024;\n  1:         12947,           79200,           132096,             32768;\n  2:        185009,         1657216,          5068800,           5636096,            1048576;\n  3:       2503611,        29601440,        132577280,         270336000,          225443840,           33554432;\n  4:      32693353,       480693312,       2841738240,        8484945920,        12976128000,         8657043456,          1073741824;\n  5:     416316835,      7323311072,      53837650944,      212183121920,       475156971520,       581330534400,        323196289024,         34359738368;\n  6:    5203074657,    106577109760,     937383817216,     4594146213888,     13579719802880,     24328036941824,      24803436134400,      11819749998592,       1099511627776;\n  7:   64093305419,   1498485501216,   15347103805440,    89988846452736,    330778527399936,    782191860645888,    1167745773207552,    1020484229529600,     425510999949312,      35184372088832;\n  8:  780480685721,  20509857734080,  239757680194560,  1637024405913600,   7199107716218880,  21169825753595904,   41716899234447360,   53382663918059520,   40819369181184000,   15129279998197760,   1125899906842624;\n  9: 9415285130163, 274729201373792, 3609734961198080, 28131567809495040, 144058147720396800, 506817183221809152, 1241963110877626368, 2097764075789352960, 2348837212394618880, 1596490883530752000, 532550655936561152, 36028797018963968;\n","recurrence_sentence":"The row polynomials satisfy (32*n+43+32*x)*P(n,x) = (352*n+825+(1024*n+2752)*x+1024*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 825 + 2752*x + 1024*x^2.","mathematica":"P[0, x_] := 825 + 2752*x + 1024*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((352*n + 825 + (1024*n + 2752)*x + 1024*x^2)*P[n - 1, x])/(32*n + 43 + 32*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 825 + 2752*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((352*n + 825 + (1024*n + 2752)*x + 1024*x^2)*v[i - 1])/(32*n + 43 + 32*x)); 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-26633b980c3508e8","steps":["basis A013611","D+1"],"edges":["e-376f2b9433b6ff6b","e-8375fa719b465100"],"text":"A038226 \u2192 basis A013611 \u2192 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