{"id":"n-431d0fbb7561f8e5","revision":0,"first_index":0,"fingerprint":"f6-1edb041cf004cf36","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= 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":false,"rows":[["1"],["5","16"],["18","104","144"],["56","456","1200","1024"],["160","1664","6336","10496","6400"],["432","5440","26880","65280","78080","36864"],["1120","16512","99840","317440","560640","522240","200704"],["2816","47488","338688","1326080","3082240","4257792","3239936","1048576"],["6912","131072","1075200","4988928","14336000","26148864","29589504","19005440","5308416"],["16640","350208","3244032","17375232","59351040","134184960","200933376","192282624","106758144","26214400"],["39424","911360","9400320","57016320","225361920","606928896","1128529920","1431306240","1185546240","579338240","126877696"],["92160","2320384","26357760","178421760","800194560","2497904640","5540806656","8737259520","9602334720","7007109120","3056599040","603979776"],["212992","5799936","71909376","537067520","2692546560","9550430208","24585437184","46298824704","63323504640","61362667520","40001077248","15753805824","2835349504"],["487424","14270464","191692800","1565065216","8668774400","34412691456","100763959296","220421160960","360321122304","434844467200","376665276416","221811572736","79607889920","13153337344"],["1105920","34635776","500957184","4437049344","26896629760","118082764800","387311468544","964455038976","1832644509696","2645053931520","2854978846720","2235127824384","1200007151616","395539644416","60397977600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","21","266","2736","25056","212976","1718496","13343616","100590336","740710656","5351256576","38053380096","267018633216","1852441767936","12725469536256"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","16"],["18","104","144"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","48","96","64"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-12","-48","-48"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204708,"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-400df3955695c308"]},{"id":204709,"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-400df3955695c308"]},{"id":204710,"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-400df3955695c308"]}],"region_exact":{},"incoming_edges":[{"id":"e-400df3955695c308","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-431d0fbb7561f8e5","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":"cb6db20df816d43eed417b00098f2b5b3cc1831a750c8374066b98afaee604e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e7d0ffa461d9dbc4","role":"sequence","name":"Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= 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:     5,     16;\n  2:    18,    104,     144;\n  3:    56,    456,    1200,     1024;\n  4:   160,   1664,    6336,    10496,     6400;\n  5:   432,   5440,   26880,    65280,    78080,     36864;\n  6:  1120,  16512,   99840,   317440,   560640,    522240,    200704;\n  7:  2816,  47488,  338688,  1326080,  3082240,   4257792,   3239936,   1048576;\n  8:  6912, 131072, 1075200,  4988928, 14336000,  26148864,  29589504,  19005440,   5308416;\n  9: 16640, 350208, 3244032, 17375232, 59351040, 134184960, 200933376, 192282624, 106758144, 26214400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+48*x+96*x^2+64*x^3)*P(n-3,x) - (12+48*x+48*x^2)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 16*x, P(2,x) = 18 + 104*x + 144*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 16*x;\nP[2, x_] := 18 + 104*x + 144*x^2;\nP[n_, x_] := P[n, x] = Expand[(8 + 48*x + 96*x^2 + 64*x^3)*P[n - 3, x] - (12 + 48*x + 48*x^2)*P[n - 2, x] + (6 + 12*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 16*x; v[3] = 18 + 104*x + 144*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 48*x + 96*x^2 + 64*x^3)*v[i - 3] - (12 + 48*x + 48*x^2)*v[i - 2] + (6 + 12*x)*v[i - 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-e7d0ffa461d9dbc4","steps":["basis A013611"],"edges":["e-400df3955695c308"],"text":"A125092 \u2192 basis A013611"},"headline":"A125092 \u2192 basis A013611","chain":null}