{"id":"n-f929f0b56bd73253","revision":0,"first_index":0,"fingerprint":"f6-4645d5c588e5d1ae","name":"basis_map_A013610 of Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["3","6"],["21","87","90"],["147","924","1935","1350"],["1029","8673","27405","38475","20250"],["7203","76146","321930","680400","718875","303750"],["50421","641067","3395700","9591750","15238125","12909375","4556250"],["352947","5243784","33385905","118077750","250543125","318937500","225534375","68343750"],["2470629","42000693","312358095","1327332825","3524968125","5990709375","6362803125","3861421875","1025156250"],["17294403","331064286","2816517060","13976701200","44584769250","94809487500","134400262500","122472000000","65097421875","15377343750"],["121060821","2576866047","24681583710","140084664300","521743902750","1332437951250","2362944150000","2873307937500","2292761953125","1084102734375","230660156250"],["847425747","19853974644","211424076675","1350816405750","5753477283750","17153224200000","36527178318750","55557317812500","59148952734375","41980148437500","17876162109375","3459902343750"],["5931980229","151689208713","1777778156385","12627075990375","60536587072500","206374728656250","512988611231250","936808899468750","1247402436328125","1181095330078125","754835361328125","292361748046875","51898535156250"],["41523861603","1150804164426","14719785225390","115056204278400","613162249363125","2352671906681250","6686541208462500","14252491464750000","22783950546328125","26978703855468750","23000277480468750","13369062656250000","4748715966796875","778478027343750"],["290667031221","8678487075027","120300559044120","1026190208329650","6017978809717875","25666137087215625","82095867059456250","200065558380187500","373275025795546875","530610185183203125","565682500195312500","438587600800781250","233776951611328125","76680085693359375","11677170410156250"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","198","4356","95832","2108304","46382688","1020419136","22449220992","493882861824","10865422960128","239039305122816","5258864712701952","115695023679442944","2545290520947744768"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","6"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-7","-15"],["7","15"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202778,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-22c9ba4d8e0c0a0e"]},{"id":202779,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-22c9ba4d8e0c0a0e"]},{"id":202780,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-22c9ba4d8e0c0a0e"]},{"id":202781,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-22c9ba4d8e0c0a0e"]}],"region_exact":{},"incoming_edges":[{"id":"e-22c9ba4d8e0c0a0e","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-f929f0b56bd73253","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"6a307cc51bc668c26cdd8e15bb40e135d55214170db0ab5d02be4a226be83de3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-cedb1276760d9bb1","role":"sequence","name":"Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.","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,         6;\n  2:       21,        87,         90;\n  3:      147,       924,       1935,        1350;\n  4:     1029,      8673,      27405,       38475,       20250;\n  5:     7203,     76146,     321930,      680400,      718875,      303750;\n  6:    50421,    641067,    3395700,     9591750,    15238125,    12909375,      4556250;\n  7:   352947,   5243784,   33385905,   118077750,   250543125,   318937500,    225534375,     68343750;\n  8:  2470629,  42000693,  312358095,  1327332825,  3524968125,  5990709375,   6362803125,   3861421875,  1025156250;\n  9: 17294403, 331064286, 2816517060, 13976701200, 44584769250, 94809487500, 134400262500, 122472000000, 65097421875, 15377343750;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(15*n-15)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (15*n - 15)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (15*n - 15)*x)*v[i - 1])/(n - 1)); 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-cedb1276760d9bb1","steps":["basis A013610"],"edges":["e-22c9ba4d8e0c0a0e"],"text":"A193726 \u2192 basis A013610"},"headline":"A193726 \u2192 basis A013610","chain":null}