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T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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":[["4","4"],["4","16","8"],["4","40","56","16"],["4","88","248","176","32"],["4","184","920","1200","512","64"],["4","376","3128","6640","4960","1408","128"],["4","760","10136","32816","38080","18368","3712","256"],["4","1528","31928","151536","256032","186368","62720","9472","512"],["4","3064","98840","670000","1583232","1630272","811776","201216","23552","1024"],["4","6136","302648","2877680","9256160","12948096","8942976","3233280","614400","57344","2048"],["4","12280","920216","12116016","52036160","96200896","88497024","43752192","11996160","1802240","137216","4096"],["4","24568","2785208","50304496","284412832","681277696","811880960","527011584","195469824","42014720","5113856","323584","8192"],["4","49144","8404760","206788400","1522673152","4656491840","7045722112","5839854592","2813251584","811086848","140281856","14110720","753664","16384"],["4","98296","25312568","843963120","8026942560","30984297344","58633038464","60810280960","36998973440","13737371648","3165274112","449892352","38019072","1736704","32768"],["4","196600","76134296","3426477616","41822639040","201959669184","472399863936","603748324608","454611322880","211371663360","62292758528","11729256448","1394032640","100352000","3964928","65536"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["8","28","116","548","2884","16644","104132","700100","5022980","38240772","307474500","2600627524","23059485060","213715233412","2064936721604"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162749,"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-9dc64aa496a51831"]}],"region_exact":{},"incoming_edges":[{"id":"e-9dc64aa496a51831","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-9e0bc9f791c813e2","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":"4a88759f67dc51933c3d81be462d488197731e54321aa20e93c56c03d7278082","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-d5fb64ec7bd8ef70","role":"sequence","name":"Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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-d5fb64ec7bd8ef70","role":"sequence","name":"Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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: 4,    4;\n  1: 4,   16,      8;\n  2: 4,   40,     56,      16;\n  3: 4,   88,    248,     176,      32;\n  4: 4,  184,    920,    1200,     512,       64;\n  5: 4,  376,   3128,    6640,    4960,     1408,     128;\n  6: 4,  760,  10136,   32816,   38080,    18368,    3712,     256;\n  7: 4, 1528,  31928,  151536,  256032,   186368,   62720,    9472,    512;\n  8: 4, 3064,  98840,  670000, 1583232,  1630272,  811776,  201216,  23552,  1024;\n  9: 4, 6136, 302648, 2877680, 9256160, 12948096, 8942976, 3233280, 614400, 57344, 2048;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 4*x.","mathematica":"P[0, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] + x*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] = 4 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] + x*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-d5fb64ec7bd8ef70","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-9dc64aa496a51831"],"text":"A227450 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A227450 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}