{"id":"n-ddb03057e82b7697","revision":1,"first_index":1,"fingerprint":"f6-cd91d972d9a103f4","name":"theta_plus_one_squared 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":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","16"],["1","40","90"],["1","72","405","320"],["1","112","1134","2240","875"],["1","160","2520","8960","8750","2016"],["1","216","4860","26880","47250","27216","4116"],["1","280","8505","67200","183750","190512","72030","7680"],["1","352","13860","147840","577500","931392","633864","168960","13365"],["1","432","21384","295680","1559250","3592512","3803184","1824768","360855","22000"],["1","520","31590","549120","3753750","11675664","17657640","13178880","4691115","715000","34606"],["1","616","45045","960960","8258250","33297264","67981914","72483840","40135095","11011000","1332331","52416"],["1","720","62370","1601600","16891875","85621536","226606380","326177280","258011325","110110000","23981958","2358720","76895"],["1","832","84240","2562560","32532500","202378176","673344672","1256386560","1341658890","817960000","277124848","49061376","3998540","109760"],["1","952","111384","3960320","59559500","445981536","1821091272","4271714304","5913237330","4866862000","2355561208","648700416","95165252","6530720","153000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","17","131","798","4362","22407","110539","529958","2487134","11480066","52287886","235558732","1051500660","4657202955","20488629195"],"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":[["-2","-18","-8"],["1","13","12"],["1","15","6"],["0","-6","-11"],["0","-4","-3"],["0","0","3"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-2","-10","14","6"],["1","17","-39","-23"],["1","27","22","36"],["0","-6","-55","3"],["0","-4","-24","14"],["0","0","3","15"],["0","0","1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-3","-1","3"],["0","2","4","4","-10"],["0","0","-20","15","5"],["0","0","-8","20","-12"],["0","0","0","11","-11"],["0","0","0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164789,"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-42aebc790a856cde"]},{"id":164790,"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-42aebc790a856cde"]},{"id":164791,"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-42aebc790a856cde"]},{"id":164792,"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-42aebc790a856cde"]}],"region_exact":{},"incoming_edges":[{"id":"e-42aebc790a856cde","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-ddb03057e82b7697","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":"56129eb58d6c0f61de253ab38483a1651a23491317a68e03de0b37496e837985","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-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":{"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: 1,  16;\n   3: 1,  40,    90;\n   4: 1,  72,   405,    320;\n   5: 1, 112,  1134,   2240,     875;\n   6: 1, 160,  2520,   8960,    8750,    2016;\n   7: 1, 216,  4860,  26880,   47250,   27216,    4116;\n   8: 1, 280,  8505,  67200,  183750,  190512,   72030,    7680;\n   9: 1, 352, 13860, 147840,  577500,  931392,  633864,  168960,  13365;\n  10: 1, 432, 21384, 295680, 1559250, 3592512, 3803184, 1824768, 360855, 22000;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n-2-(4*n^4+6*n^3-15*n^2-13*n+18)*x+(n^6+3*n^5-3*n^4-11*n^3+6*n^2+12*n-8)*x^2)*P(n,x) = (n^2+n-2-(4*n^4+6*n^3-27*n^2-17*n+10)*x+(n^6+3*n^5-24*n^4-55*n^3+22*n^2-39*n+14)*x^2+(3*n^6+15*n^5+14*n^4+3*n^3+36*n^2-23*n+6)*x^3)*P(n-1,x) + ((2*n+1)*x-(8*n^3+20*n^2-4*n+3)*x^2+(2*n^5+11*n^4+20*n^3+15*n^2+4*n-1)*x^3-(2*n^5+11*n^4+12*n^3-5*n^2+10*n-3)*x^4)*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[((n^2 + n - 2 - (4*n^4 + 6*n^3 - 27*n^2 - 17*n + 10)*x + (n^6 + 3*n^5 - 24*n^4 - 55*n^3 + 22*n^2 - 39*n + 14)*x^2 + (3*n^6 + 15*n^5 + 14*n^4 + 3*n^3 + 36*n^2 - 23*n + 6)*x^3)*P[n - 1, x] + ((2*n + 1)*x - (8*n^3 + 20*n^2 - 4*n + 3)*x^2 + (2*n^5 + 11*n^4 + 20*n^3 + 15*n^2 + 4*n - 1)*x^3 - (2*n^5 + 11*n^4 + 12*n^3 - 5*n^2 + 10*n - 3)*x^4)*D[P[n - 1, x], x])/(n^2 + n - 2 - (4*n^4 + 6*n^3 - 15*n^2 - 13*n + 18)*x + (n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 6*n^2 + 12*n - 8)*x^2)]];\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] = ((n^2 + n - 2 - (4*n^4 + 6*n^3 - 27*n^2 - 17*n + 10)*x + (n^6 + 3*n^5 - 24*n^4 - 55*n^3 + 22*n^2 - 39*n + 14)*x^2 + (3*n^6 + 15*n^5 + 14*n^4 + 3*n^3 + 36*n^2 - 23*n + 6)*x^3)*v[i - 1] + ((2*n + 1)*x - (8*n^3 + 20*n^2 - 4*n + 3)*x^2 + (2*n^5 + 11*n^4 + 20*n^3 + 15*n^2 + 4*n - 1)*x^3 - (2*n^5 + 11*n^4 + 12*n^3 - 5*n^2 + 10*n - 3)*x^4)*deriv(v[i - 1], x))/(n^2 + n - 2 - (4*n^4 + 6*n^3 - 15*n^2 - 13*n + 18)*x + (n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 6*n^2 + 12*n - 8)*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":["(\u03b8+1)\u00b2"],"edges":["e-42aebc790a856cde"],"text":"A062145 \u2192 (\u03b8+1)\u00b2"},"headline":"A062145 \u2192 (\u03b8+1)\u00b2","chain":null}