{"id":"n-0e6cb9175abc81e3","revision":1,"first_index":0,"fingerprint":"f6-8c5ce1bab599866b","name":"obreschkoff_combination of Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2","1"],["2","13","1"],["2","95","84","1"],["2","597","1909","515","1"],["2","3619","30136","28824","3106","1"],["2","21761","404925","934230","379720","18657","1"],["2","130623","5004436","23309555","22780250","4646477","111968","1"],["2","783805","59004829","501185141","963356305","483409327","54488975","671839","1"],["2","4702907","676710264","9824972850","33149277896","32687106732","9400568352","623005094","4031070","1"],["2","28217529","7633272901","181289250064","998315086636","1707041236978","975452952754","172565391976","7014970714","24186461","1"],["2","169305271","85243852900","3210480523685","27444298864940","75169798221014","74461072070672","26650135162250","3045548404610","78257099715","145118812","1"],["2","1015831733","946196018421","55253850914135","707100262341225","2937578979077154","4630182814483506","2890562205871422","683660607702660","52293546362425","868108223977","870712923","1"],["2","6094990515","10464317185624","931926446999356","17375589963891110","105235773251478579","248921035669219920","247228431833813568","103052311307948382","16747183207102585","880579096655352","9597224793380","5224277594","1"],["2","36569943217","115474929898669","15492485098858810","412154561471434616","3532004132526245639","12016898388987599799","17838597390863334348","11874905730405264810","3446327411182188367","396339304711331387","14622370319389082","105883800095804","31345665625","1"],["2","219419659439","1272611723749716","254875750535157479","9518426400768469366","112768515241257284955","534484181708634339728","1133660518280471399907","1127838819278595665358","526141280110755320557","109734342391546531204","9137579581155996789","240368855145667066","1166764493597129","188073993816","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","16","182","3024","65688","1759296","55983312","2062900224","86370375168","4049364566016","210145148623872","11958448457539584","740392903638355968","49547563202211250176","3563781347382576832512"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","1"],["2","13","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-9"],["0","5"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","5","-5"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["0","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5","-5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162842,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-45037657f8d01e76"]}],"region_exact":{},"incoming_edges":[{"id":"e-45037657f8d01e76","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-0e6cb9175abc81e3","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"eab889b1ef6a0f222c20b1f798feafd5f683d8450cbc8fc2b09b6ddad2a7517b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-05516f61aeaea779","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-05516f61aeaea779","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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,        1;\n  1: 2,       13,          1;\n  2: 2,       95,         84,            1;\n  3: 2,      597,       1909,          515,            1;\n  4: 2,     3619,      30136,        28824,         3106,             1;\n  5: 2,    21761,     404925,       934230,       379720,         18657,            1;\n  6: 2,   130623,    5004436,     23309555,     22780250,       4646477,       111968,            1;\n  7: 2,   783805,   59004829,    501185141,    963356305,     483409327,     54488975,       671839,          1;\n  8: 2,  4702907,  676710264,   9824972850,  33149277896,   32687106732,   9400568352,    623005094,    4031070,        1;\n  9: 2, 28217529, 7633272901, 181289250064, 998315086636, 1707041236978, 975452952754, 172565391976, 7014970714, 24186461, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(5*n-9)*x)*P(n-2,x) + (5*x-5*x^2)*d/dx P(n-2,x) + (5*n+1)*x*P(n-1,x) + (5*x-5*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 2 + 13*x + x^2.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 2 + 13*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (5*n - 9)*x)*P[n - 2, x] + (5*x - 5*x^2)*D[P[n - 2, x], x] + (5*n + 1)*x*P[n - 1, x] + (5*x - 5*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; v[2] = 2 + 13*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (5*n - 9)*x)*v[i - 2] + (5*x - 5*x^2)*deriv(v[i - 2], x) + (5*n + 1)*x*v[i - 1] + (5*x - 5*x^2)*deriv(v[i - 1], 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-05516f61aeaea779","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-45037657f8d01e76"],"text":"A142460 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A142460 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}