{"id":"n-3221e64094f10734","revision":0,"first_index":1,"fingerprint":"f6-e9dc029a14cf0a23","name":"theta_plus_one_squared of basis_map_A007318 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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":[["1"],["4","4"],["11","28","9"],["29","124","90","16"],["76","460","540","208","25"],["199","1560","2565","1568","400","36"],["521","5016","10665","9056","3625","684","49"],["1364","15572","40716","44560","24625","7236","1078","64"],["3571","47156","146520","197008","139875","56484","13034","1600","81"],["9349","140180","504945","806944","702825","363636","114905","21760","2268","100"],["24476","410780","1683720","3121504","3229450","2046492","826630","213760","34263","3100","121"],["64079","1190064","5470470","11550848","13862875","10426248","5150488","1699200","371061","51500","4114","144"],["167761","3415728","17405334","41256320","56407375","49194792","28816116","11611008","3229470","609500","74536","5328","169"],["439204","9728164","54430920","143160928","219822250","218384748","148257438","70771200","24012531","5764000","956989","104544","6760","196"],["1149851","27525580","167775795","484992544","826748325","922503492","713202105","394344960","158377923","46327600","9770871","1447200","142805","8428","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","48","259","1309","6328","29616","135215","605329","2666912","11594296","49841091","212193437","895839872","3754317704"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-300","-120","312","120","-12"],["200","80","-658","-242","44"],["0","0","450","162","-63"],["0","0","-100","-36","44"],["0","0","0","0","-15"],["0","0","0","0","2"]]},"terms":[{"coeff":{"coeffs":[["-200","-130","48","46","8","0"],["300","-5","-452","-279","-44","0"],["0","175","550","318","34","-4"],["0","0","-150","-79","19","8"],["0","0","0","-10","-20","-5"],["0","0","0","0","3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["125","300","160","-64","-57","-8","0"],["0","0","350","600","286","36","0"],["0","0","-175","-300","-123","6","4"],["0","0","0","0","-20","-24","-4"],["0","0","0","0","5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175459,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-45a3ddb230a79937"]},{"id":175460,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-45a3ddb230a79937"]},{"id":175461,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-45a3ddb230a79937"]},{"id":175462,"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-45a3ddb230a79937"]}],"region_exact":{},"incoming_edges":[{"id":"e-45a3ddb230a79937","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-3221e64094f10734","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"676f6229b9c97dd0bc65d5d6b4f5196af9bcdd416539b6dd9a71c71ccc4beaa2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-67d789c805b2e2cd","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    4,      4;\n   3:   11,     28,      9;\n   4:   29,    124,     90,     16;\n   5:   76,    460,    540,    208,     25;\n   6:  199,   1560,   2565,   1568,    400,     36;\n   7:  521,   5016,  10665,   9056,   3625,    684,     49;\n   8: 1364,  15572,  40716,  44560,  24625,   7236,   1078,    64;\n   9: 3571,  47156, 146520, 197008, 139875,  56484,  13034,  1600,   81;\n  10: 9349, 140180, 504945, 806944, 702825, 363636, 114905, 21760, 2268, 100;\n","recurrence_sentence":"The row polynomials satisfy (200*n-300+(80*n-120)*x-(100*n^3-450*n^2+658*n-312)*x^2-(36*n^3-162*n^2+242*n-120)*x^3+(2*n^5-15*n^4+44*n^3-63*n^2+44*n-12)*x^4)*P(n,x) = (300*n-200+(175*n^2-5*n-130)*x-(150*n^3-550*n^2+452*n-48)*x^2-(10*n^4+79*n^3-318*n^2+279*n-46)*x^3+(3*n^5-20*n^4+19*n^3+34*n^2-44*n+8)*x^4+(n^5-5*n^4+8*n^3-4*n^2)*x^5)*P(n-1,x) + (125+300*x-(175*n^2-350*n-160)*x^2-(300*n^2-600*n+64)*x^3+(5*n^4-20*n^3-123*n^2+286*n-57)*x^4+(6*n^4-24*n^3+6*n^2+36*n-8)*x^5+(n^4-4*n^3+4*n^2)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((300*n - 200 + (175*n^2 - 5*n - 130)*x - (150*n^3 - 550*n^2 + 452*n - 48)*x^2 - (10*n^4 + 79*n^3 - 318*n^2 + 279*n - 46)*x^3 + (3*n^5 - 20*n^4 + 19*n^3 + 34*n^2 - 44*n + 8)*x^4 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^5)*P[n - 1, x] + (125 + 300*x - (175*n^2 - 350*n - 160)*x^2 - (300*n^2 - 600*n + 64)*x^3 + (5*n^4 - 20*n^3 - 123*n^2 + 286*n - 57)*x^4 + (6*n^4 - 24*n^3 + 6*n^2 + 36*n - 8)*x^5 + (n^4 - 4*n^3 + 4*n^2)*x^6)*D[P[n - 1, x], x])/(200*n - 300 + (80*n - 120)*x - (100*n^3 - 450*n^2 + 658*n - 312)*x^2 - (36*n^3 - 162*n^2 + 242*n - 120)*x^3 + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((300*n - 200 + (175*n^2 - 5*n - 130)*x - (150*n^3 - 550*n^2 + 452*n - 48)*x^2 - (10*n^4 + 79*n^3 - 318*n^2 + 279*n - 46)*x^3 + (3*n^5 - 20*n^4 + 19*n^3 + 34*n^2 - 44*n + 8)*x^4 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^5)*v[i - 1] + (125 + 300*x - (175*n^2 - 350*n - 160)*x^2 - (300*n^2 - 600*n + 64)*x^3 + (5*n^4 - 20*n^3 - 123*n^2 + 286*n - 57)*x^4 + (6*n^4 - 24*n^3 + 6*n^2 + 36*n - 8)*x^5 + (n^4 - 4*n^3 + 4*n^2)*x^6)*deriv(v[i - 1], x))/(200*n - 300 + (80*n - 120)*x - (100*n^3 - 450*n^2 + 658*n - 312)*x^2 - (36*n^3 - 162*n^2 + 242*n - 120)*x^3 + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*x^4)); 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-318700a428d7b97b","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-0092e3a0a8732095","e-45a3ddb230a79937"],"text":"A111125 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A111125 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}