{"id":"n-6aab2c08337784b8","revision":0,"first_index":0,"fingerprint":"f6-fc9cd9a0d3c1ee91","name":"basis_map_A013610 of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.","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":[["2"],["26","39"],["338","1014","18"],["4394","19773","19809","54"],["57122","342732","536517","67257","162"],["742586","5569395","13416534","10249848","225261","486"],["9653618","86882562","267295212","300532842","60076485","746253","1458"],["125497034","1317718857","5103072927","8624372670","5451148611","278428428","2450169","4374"],["1631461442","19577537304","89655136839","190203806475","174294941805","42789828933","1160536653","7984737","13122"],["21208998746","286321483071","1533083703804","4071876600960","5398571429838","2986822502604","250486729884","4549327416","25856901","39366"],["275716983698","4135754755470","25144579928214","78404202343188","129534382318446","103060834866486","27984234321216","1259570886012","17127024669","83278773","118098"],["3584320788074","59141293003221","404759310004269","1472213407340214","3013112147945130","3337827577961184","1682317429232418","194581205676396","5768325705681","62645871420","266960529","354294"],["46596170244962","838731064409316","6356856218849433","26157530468972925","62601683994676332","86022941000278590","61592961258004770","17621469859551426","1138085343067365","24789671307423","224215016949","852254217","1062882"],["605750213184506","11812129157097867","98482101659919090","455553890835976152","1266130548301376835","2129785389101496258","2053754168842429020","969561113045549112","139440788075875275","5947405459167150","101730698150994","788926290264","2710880541","3188646"],["7874752771398578","165369808199370138","1501246656718197264","7674172261601244606","24072544151538782427","47215365419863107819","56208499654003544034","37043635302218285532","10909921689094592835","922000710318821199","28712509512111612","403080061657662","2737969683093","8594995293","9565938"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","65","1370","44030","1003790","30204110","725188430","20902693070","519321247310","14552946672590","369816486824270","10173367930842830","262401870117696590","7131175807030581710","185749748742475358030"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["26","39"],["338","1014","18"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-286","-858"],["143","429"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","-6"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203569,"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-537a2c184630f963"]},{"id":203570,"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-537a2c184630f963"]},{"id":203571,"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-537a2c184630f963"]},{"id":203572,"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-537a2c184630f963"]}],"region_exact":{},"incoming_edges":[{"id":"e-537a2c184630f963","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-6aab2c08337784b8","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":"366ec392af4417ef165da4d7b5f3b46a52a734e0867b5847afd959cecac7013f","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-2a67b0774691ba3b","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.","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:           2;\n  1:          26,           39;\n  2:         338,         1014,            18;\n  3:        4394,        19773,         19809,            54;\n  4:       57122,       342732,        536517,         67257,           162;\n  5:      742586,      5569395,      13416534,      10249848,        225261,           486;\n  6:     9653618,     86882562,     267295212,     300532842,      60076485,        746253,         1458;\n  7:   125497034,   1317718857,    5103072927,    8624372670,    5451148611,     278428428,      2450169,       4374;\n  8:  1631461442,  19577537304,   89655136839,  190203806475,  174294941805,   42789828933,   1160536653,    7984737,    13122;\n  9: 21208998746, 286321483071, 1533083703804, 4071876600960, 5398571429838, 2986822502604, 250486729884, 4549327416, 25856901, 39366;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (143*n-286+(429*n-858)*x)*P(n-2,x) + (2*n-4+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 26 + 39*x, P(2,x) = 338 + 1014*x + 18*x^2.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 26 + 39*x;\nP[2, x_] := 338 + 1014*x + 18*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((143*n - 286 + (429*n - 858)*x)*P[n - 2, x] + (2*n - 4 + (3*n - 6)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 26 + 39*x; v[3] = 338 + 1014*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((143*n - 286 + (429*n - 858)*x)*v[i - 2] + (2*n - 4 + (3*n - 6)*x)*v[i - 1])/(n - 2)); 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-2a67b0774691ba3b","steps":["basis A013610"],"edges":["e-537a2c184630f963"],"text":"A153651 \u2192 basis A013610"},"headline":"A153651 \u2192 basis A013610","chain":null}
