{"id":"n-8b65fbf19b8d0732","revision":0,"first_index":1,"fingerprint":"f6-c3466227e6835c89","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.","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","4"],["1","16","27"],["1","36","162","160"],["1","64","540","1280","875"],["1","100","1350","5600","8750","4536"],["1","144","2835","17920","47250","54432","22638"],["1","196","5292","47040","183750","349272","316932","109824"],["1","256","9072","107520","577500","1596672","2354352","1757184","521235"],["1","324","14580","221760","1559250","5837832","12360348","14826240","9382230","2431000"],["1","400","22275","422400","3753750","18162144","51501450","87859200","88609950","48620000","11177738"],["1","484","32670","755040","8258250","49945896","181285104","410741760","584825670","508079000","245910236","50791104"],["1","576","46332","1281280","16891875","124540416","560335776","1613094912","3030460290","3695120000","2816789976","1218986496","228501182"],["1","676","63882","2082080","32532500","286702416","1560935376","5533811712","13131994590","21015995000","22377831476","15186540096","5941030732","1019258800"],["1","784","85995","3261440","59559500","617512896","3992392404","17027112960","49497518070","99583484000","138570417986","130837883904","79975413700","28539246400","4513117500"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","44","359","2760","20337","145220","1012307","6923792","46633565","310129308","2040625215","13306049112","86088779337","553217007540"],"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"],["1","4"],["1","16","27"],["1","36","162","160"]],"recurrence":{"denominator":{"coeffs":[["126"],["-253"],["167"],["-44"],["4"]]},"terms":[{"coeff":{"coeffs":[["-300"],["455"],["-239"],["52"],["-4"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1056","4320"],["-1640","-6648"],["888","3568"],["-200","-800"],["16","64"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4320"],["0","0","6648"],["0","0","-3568"],["0","0","800"],["0","0","-64"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1350","-7344","34020"],["2172","11712","-53568"],["-1226","-6560","29664"],["288","1536","-6912"],["-24","-128","576"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","7344","15120"],["0","0","-11712","-23808"],["0","0","6560","13184"],["0","0","-1536","-3072"],["0","0","128","256"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["720","3024"],["-1240","-5064"],["744","2992"],["-184","-736"],["16","64"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-3024"],["0","0","5064"],["0","0","-2992"],["0","0","736"],["0","0","-64"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177301,"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-c40363197fc0342f"]},{"id":177302,"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-c40363197fc0342f"]},{"id":177303,"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-c40363197fc0342f"]}],"region_exact":{},"incoming_edges":[{"id":"e-c40363197fc0342f","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-8b65fbf19b8d0732","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":"8ce3166c64c84869bf390512f8536e15b09027d61f9de03a96c3d19b443344b3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-55697ddae4d22d60","role":"sequence","name":"Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.","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,   4;\n   3: 1,  16,    27;\n   4: 1,  36,   162,    160;\n   5: 1,  64,   540,   1280,     875;\n   6: 1, 100,  1350,   5600,    8750,    4536;\n   7: 1, 144,  2835,  17920,   47250,   54432,    22638;\n   8: 1, 196,  5292,  47040,  183750,  349272,   316932,   109824;\n   9: 1, 256,  9072, 107520,  577500, 1596672,  2354352,  1757184,  521235;\n  10: 1, 324, 14580, 221760, 1559250, 5837832, 12360348, 14826240, 9382230, 2431000;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+888*n^2-1640*n+1056+(64*n^4-800*n^3+3568*n^2-6648*n+4320)*x)*P(n-3,x) - (64*n^4-800*n^3+3568*n^2-6648*n+4320)*x^2*d/dx P(n-3,x) - (24*n^4-288*n^3+1226*n^2-2172*n+1350+(128*n^4-1536*n^3+6560*n^2-11712*n+7344)*x-(576*n^4-6912*n^3+29664*n^2-53568*n+34020)*x^2)*P(n-2,x) + ((128*n^4-1536*n^3+6560*n^2-11712*n+7344)*x^2+(256*n^4-3072*n^3+13184*n^2-23808*n+15120)*x^3)*d/dx P(n-2,x) + (16*n^4-184*n^3+744*n^2-1240*n+720+(64*n^4-736*n^3+2992*n^2-5064*n+3024)*x)*P(n-1,x) - (64*n^4-736*n^3+2992*n^2-5064*n+3024)*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 1 + 16*x + 27*x^2, P(4,x) = 1 + 36*x + 162*x^2 + 160*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[3, x_] := 1 + 16*x + 27*x^2;\nP[4, x_] := 1 + 36*x + 162*x^2 + 160*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 888*n^2 - 1640*n + 1056 + (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x)*P[n - 3, x] - (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x^2*D[P[n - 3, x], x] - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (128*n^4 - 1536*n^3 + 6560*n^2 - 11712*n + 7344)*x - (576*n^4 - 6912*n^3 + 29664*n^2 - 53568*n + 34020)*x^2)*P[n - 2, x] + ((128*n^4 - 1536*n^3 + 6560*n^2 - 11712*n + 7344)*x^2 + (256*n^4 - 3072*n^3 + 13184*n^2 - 23808*n + 15120)*x^3)*D[P[n - 2, x], x] + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x)*P[n - 1, x] - (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x^2*D[P[n - 1, x], x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 16*x + 27*x^2; v[4] = 1 + 36*x + 162*x^2 + 160*x^3; for(n = 5, N, my(i = n); v[i] = (-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*v[i - 4] + (16*n^4 - 200*n^3 + 888*n^2 - 1640*n + 1056 + (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x)*v[i - 3] - (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x^2*deriv(v[i - 3], x) - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (128*n^4 - 1536*n^3 + 6560*n^2 - 11712*n + 7344)*x - (576*n^4 - 6912*n^3 + 29664*n^2 - 53568*n + 34020)*x^2)*v[i - 2] + ((128*n^4 - 1536*n^3 + 6560*n^2 - 11712*n + 7344)*x^2 + (256*n^4 - 3072*n^3 + 13184*n^2 - 23808*n + 15120)*x^3)*deriv(v[i - 2], x) + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x)*v[i - 1] - (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x^2*deriv(v[i - 1], x))/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)); 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-55697ddae4d22d60","steps":["(\u03b8+1)\u00b2"],"edges":["e-c40363197fc0342f"],"text":"A123160 \u2192 (\u03b8+1)\u00b2"},"headline":"A123160 \u2192 (\u03b8+1)\u00b2","chain":null}