{"id":"n-b7ab2dc5f7577856","revision":1,"first_index":1,"fingerprint":"f6-27b0ef728bca3ff5","name":"obreschkoff_combination of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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","4"],["2","14","10"],["2","28","55","20"],["2","46","171","160","35"],["2","68","406","700","385","56"],["2","94","820","2240","2240","812","84"],["2","124","1485","5880","9240","6048","1554","120"],["2","158","2485","13440","30450","31164","14406","2760","165"],["2","196","3916","27720","85470","125664","90552","31152","4620","220"],["2","238","5886","52800","212520","424116","437976","234432","62370","7370","286"],["2","284","8515","94380","480480","1249248","1747746","1338480","553410","117260","11297","364"],["2","334","11935","160160","1006005","3303300","6012006","6229080","3680820","1211210","209209","16744","455"],["2","388","16290","260260","1976975","7999992","18366348","24727560","19749015","9280700","2488486","357084","24115","560"],["2","446","21736","407680","3683680","18009992","50906856","86376576","89566620","56848220","21757736","4845568","586768","33880","680"],["2","508","28441","618800","6559280","38118080","130077948","271532976","354585660","292011720","150872722","47944624","9009728","932960","46580","816"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["6","26","105","414","1617","6292","24453","95030","369512","1437996","5601466","21841260","85247775","333046440","1302340845"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","4"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-24","24","-6"],["4","2","1"],["16","-2","4"],["4","0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-24","36","-12"],["-24","8","36","-8"],["4","18","21","7"],["16","34","2","10"],["4","12","1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-12","36","-30","6"],["0","4","18","-23","1"],["0","24","-22","3","-5"],["0","8","-8","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162724,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-a6e517cc00955d8a"]}],"region_exact":{},"incoming_edges":[{"id":"e-a6e517cc00955d8a","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-b7ab2dc5f7577856","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":"35bde596e1c0e98e25b1d0a563b4462769b599db232c9363bf29337d025f460e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-a2304b34e998691c","role":"sequence","name":"Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-a2304b34e998691c","role":"sequence","name":"Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 2,   4;\n   2: 2,  14,   10;\n   3: 2,  28,   55,    20;\n   4: 2,  46,  171,   160,     35;\n   5: 2,  68,  406,   700,    385,     56;\n   6: 2,  94,  820,  2240,   2240,    812,     84;\n   7: 2, 124, 1485,  5880,   9240,   6048,   1554,    120;\n   8: 2, 158, 2485, 13440,  30450,  31164,  14406,   2760,   165;\n   9: 2, 196, 3916, 27720,  85470, 125664,  90552,  31152,  4620,  220;\n  10: 2, 238, 5886, 52800, 212520, 424116, 437976, 234432, 62370, 7370, 286;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4+16*n^3+4*n^2-24*n-(2*n^3-2*n^2-24*n)*x+(n^4+4*n^3+n^2-6*n)*x^2)*P(n,x) = (4*n^4+16*n^3+4*n^2-24*n+(12*n^4+34*n^3+18*n^2+8*n-24)*x+(n^4+2*n^3+21*n^2+36*n+36)*x^2+(3*n^4+10*n^3+7*n^2-8*n-12)*x^3)*P(n-1,x) + ((8*n^3+24*n^2+4*n-12)*x-(8*n^3+22*n^2-18*n-36)*x^2+(2*n^3+3*n^2-23*n-30)*x^3-(2*n^3+5*n^2-n-6)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2 + 4*x.","mathematica":"P[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^4 + 16*n^3 + 4*n^2 - 24*n + (12*n^4 + 34*n^3 + 18*n^2 + 8*n - 24)*x + (n^4 + 2*n^3 + 21*n^2 + 36*n + 36)*x^2 + (3*n^4 + 10*n^3 + 7*n^2 - 8*n - 12)*x^3)*P[n - 1, x] + ((8*n^3 + 24*n^2 + 4*n - 12)*x - (8*n^3 + 22*n^2 - 18*n - 36)*x^2 + (2*n^3 + 3*n^2 - 23*n - 30)*x^3 - (2*n^3 + 5*n^2 - n - 6)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 16*n^3 + 4*n^2 - 24*n - (2*n^3 - 2*n^2 - 24*n)*x + (n^4 + 4*n^3 + n^2 - 6*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + 4*x; for(n = 2, N, my(i = n); v[i] = ((4*n^4 + 16*n^3 + 4*n^2 - 24*n + (12*n^4 + 34*n^3 + 18*n^2 + 8*n - 24)*x + (n^4 + 2*n^3 + 21*n^2 + 36*n + 36)*x^2 + (3*n^4 + 10*n^3 + 7*n^2 - 8*n - 12)*x^3)*v[i - 1] + ((8*n^3 + 24*n^2 + 4*n - 12)*x - (8*n^3 + 22*n^2 - 18*n - 36)*x^2 + (2*n^3 + 3*n^2 - 23*n - 30)*x^3 - (2*n^3 + 5*n^2 - n - 6)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 16*n^3 + 4*n^2 - 24*n - (2*n^3 - 2*n^2 - 24*n)*x + (n^4 + 4*n^3 + n^2 - 6*n)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a2304b34e998691c","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a6e517cc00955d8a"],"text":"A062145 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A062145 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}