{"id":"n-86c57fc08f195991","revision":0,"first_index":0,"fingerprint":"f6-c5e5ab6ef10668e5","name":"theta_plus_one_squared of Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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":[["1"],["1","4"],["1","68","9"],["1","472","1062","16"],["1","2916","24129","11664","25"],["1","17600","375705","667920","110000","36"],["1","105724","4997160","21587920","13881000","951516","49"],["1","634488","61351614","535306960","836417125","245406456","7772478","64"],["1","3807092","720312687","11445138496","35298174875","25751561616","3921702407","60913472","81"],["1","22842736","8238951999","223364796144","1210402509400","1742979613536","684054688191","58588103104","462565404","100"],["1","137056620","92778544170","4107532445680","36327100155125","90889086762972","71201116304045","16430129782720","835006897530","3426415500","121"],["1","822339944","1035006326820","72556279257040","995662840797000","3993472904829732","5435560342684913","2548896872440320","367316163738765","11500070298000","24875783306","144"],["1","4934039908","11480974990569","1246365683676320","25589146393873000","155707012313606412","337623291975823721","276812466335300288","82908834316148520","7789785522977000","154355330428761","177625436688","169"],["1","29604239712","126921665249037","20991992727537776","627521453717264125","5566334295701767032","18123143456330674104","23671044514391084544","12524252165328975822","2510085814869056500","158751945002004431","2030746643984592","1250779127832","196"],["1","177625438556","1400257007588844","348609693647535824","14860487180441934400","186475954636513404072","873509497648519884397","1706520091840898160448","1443964679786328788493","517988762879203900200","71924757953338962496","3137487242827822416","26293714920279404","8703646489244","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","78","1551","38735","1171262","41523370","1686889186","77201610727","3928114070615","219886314364484","13426026178495985","887842921406301357","63204284586760965704","4818758031714920189020"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4"],["1","68","9"],["1","472","1062","16"]],"recurrence":{"denominator":{"coeffs":[["441"],["-840"],["400"]]},"terms":[{"coeff":{"coeffs":[["0","0","30975"],["0","0","-99000"],["0","0","90000"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","15225"],["0","0","0","-44000"],["0","0","0","40000"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","41580","-88935"],["0","8800","411600"],["0","-8000","-542000"],["0","0","180000"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","13545","-91035"],["0","0","-8800","329725"],["0","0","8000","-372000"],["0","0","0","80000"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","15225","-15225"],["0","0","0","-44000","44000"],["0","0","0","40000","-40000"]]},"deriv_order":2,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["84","-66024","3969"],["-1760","8940","-188370"],["1600","31200","447225"],["0","-8000","-353000"],["0","0","90000"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","12390","-59556","8379"],["0","4400","-19740","-240870"],["0","-4000","16800","480100"],["0","0","8000","-288000"],["0","0","0","40000"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","9975","-12180","2205"],["0","0","11000","37300","-48300"],["0","0","-10000","-76000","86000"],["0","0","0","40000","-40000"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["357","-7056"],["920","22260"],["-1200","-23200"],["0","8000"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","210","-1764"],["0","-6400","-5460"],["0","4000","15200"],["0","0","-8000"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189431,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-4aa333fe2966d42c"]},{"id":189432,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-4aa333fe2966d42c"]},{"id":189433,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-4aa333fe2966d42c"]},{"id":189434,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-4aa333fe2966d42c"]}],"region_exact":{},"incoming_edges":[{"id":"e-4aa333fe2966d42c","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-86c57fc08f195991","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":"17a35e6f3642e8626110b6450fb5ce7918cf30c7364c1d734a9956ca99c86e14","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-aac530d688cc3150","role":"sequence","name":"Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,        4;\n  2: 1,       68,          9;\n  3: 1,      472,       1062,           16;\n  4: 1,     2916,      24129,        11664,            25;\n  5: 1,    17600,     375705,       667920,        110000,            36;\n  6: 1,   105724,    4997160,     21587920,      13881000,        951516,           49;\n  7: 1,   634488,   61351614,    535306960,     836417125,     245406456,      7772478,          64;\n  8: 1,  3807092,  720312687,  11445138496,   35298174875,   25751561616,   3921702407,    60913472,        81;\n  9: 1, 22842736, 8238951999, 223364796144, 1210402509400, 1742979613536, 684054688191, 58588103104, 462565404, 100;\n","recurrence_sentence":"The row polynomials satisfy (400*n^2-840*n+441)*P(n,x) = (90000*n^2-99000*n+30975)*x^2*P(n-4,x) + (40000*n^2-44000*n+15225)*x^3*d/dx P(n-4,x) - ((8000*n^2-8800*n-41580)*x-(180000*n^3-542000*n^2+411600*n-88935)*x^2)*P(n-3,x) + ((8000*n^2-8800*n+13545)*x^2+(80000*n^3-372000*n^2+329725*n-91035)*x^3)*d/dx P(n-3,x) + ((40000*n^2-44000*n+15225)*x^3-(40000*n^2-44000*n+15225)*x^4)*d^2/dx^2 P(n-3,x) + (1600*n^2-1760*n+84-(8000*n^3-31200*n^2-8940*n+66024)*x+(90000*n^4-353000*n^3+447225*n^2-188370*n+3969)*x^2)*P(n-2,x) - ((4000*n^2-4400*n-12390)*x-(8000*n^3+16800*n^2-19740*n-59556)*x^2-(40000*n^4-288000*n^3+480100*n^2-240870*n+8379)*x^3)*d/dx P(n-2,x) - ((10000*n^2-11000*n-9975)*x^2-(40000*n^3-76000*n^2+37300*n-12180)*x^3+(40000*n^3-86000*n^2+48300*n-2205)*x^4)*d^2/dx^2 P(n-2,x) - (1200*n^2-920*n-357-(8000*n^3-23200*n^2+22260*n-7056)*x)*P(n-1,x) + ((4000*n^2-6400*n+210)*x-(8000*n^3-15200*n^2+5460*n+1764)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 68*x + 9*x^2, P(3,x) = 1 + 472*x + 1062*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[2, x_] := 1 + 68*x + 9*x^2;\nP[3, x_] := 1 + 472*x + 1062*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((90000*n^2 - 99000*n + 30975)*x^2*P[n - 4, x] + (40000*n^2 - 44000*n + 15225)*x^3*D[P[n - 4, x], x] - ((8000*n^2 - 8800*n - 41580)*x - (180000*n^3 - 542000*n^2 + 411600*n - 88935)*x^2)*P[n - 3, x] + ((8000*n^2 - 8800*n + 13545)*x^2 + (80000*n^3 - 372000*n^2 + 329725*n - 91035)*x^3)*D[P[n - 3, x], x] + ((40000*n^2 - 44000*n + 15225)*x^3 - (40000*n^2 - 44000*n + 15225)*x^4)*D[P[n - 3, x], {x, 2}] + (1600*n^2 - 1760*n + 84 - (8000*n^3 - 31200*n^2 - 8940*n + 66024)*x + (90000*n^4 - 353000*n^3 + 447225*n^2 - 188370*n + 3969)*x^2)*P[n - 2, x] - ((4000*n^2 - 4400*n - 12390)*x - (8000*n^3 + 16800*n^2 - 19740*n - 59556)*x^2 - (40000*n^4 - 288000*n^3 + 480100*n^2 - 240870*n + 8379)*x^3)*D[P[n - 2, x], x] - ((10000*n^2 - 11000*n - 9975)*x^2 - (40000*n^3 - 76000*n^2 + 37300*n - 12180)*x^3 + (40000*n^3 - 86000*n^2 + 48300*n - 2205)*x^4)*D[P[n - 2, x], {x, 2}] - (1200*n^2 - 920*n - 357 - (8000*n^3 - 23200*n^2 + 22260*n - 7056)*x)*P[n - 1, x] + ((4000*n^2 - 6400*n + 210)*x - (8000*n^3 - 15200*n^2 + 5460*n + 1764)*x^2)*D[P[n - 1, x], x])/(400*n^2 - 840*n + 441)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 68*x + 9*x^2; v[4] = 1 + 472*x + 1062*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((90000*n^2 - 99000*n + 30975)*x^2*v[i - 4] + (40000*n^2 - 44000*n + 15225)*x^3*deriv(v[i - 4], x) - ((8000*n^2 - 8800*n - 41580)*x - (180000*n^3 - 542000*n^2 + 411600*n - 88935)*x^2)*v[i - 3] + ((8000*n^2 - 8800*n + 13545)*x^2 + (80000*n^3 - 372000*n^2 + 329725*n - 91035)*x^3)*deriv(v[i - 3], x) + ((40000*n^2 - 44000*n + 15225)*x^3 - (40000*n^2 - 44000*n + 15225)*x^4)*deriv(deriv(v[i - 3], x), x) + (1600*n^2 - 1760*n + 84 - (8000*n^3 - 31200*n^2 - 8940*n + 66024)*x + (90000*n^4 - 353000*n^3 + 447225*n^2 - 188370*n + 3969)*x^2)*v[i - 2] - ((4000*n^2 - 4400*n - 12390)*x - (8000*n^3 + 16800*n^2 - 19740*n - 59556)*x^2 - (40000*n^4 - 288000*n^3 + 480100*n^2 - 240870*n + 8379)*x^3)*deriv(v[i - 2], x) - ((10000*n^2 - 11000*n - 9975)*x^2 - (40000*n^3 - 76000*n^2 + 37300*n - 12180)*x^3 + (40000*n^3 - 86000*n^2 + 48300*n - 2205)*x^4)*deriv(deriv(v[i - 2], x), x) - (1200*n^2 - 920*n - 357 - (8000*n^3 - 23200*n^2 + 22260*n - 7056)*x)*v[i - 1] + ((4000*n^2 - 6400*n + 210)*x - (8000*n^3 - 15200*n^2 + 5460*n + 1764)*x^2)*deriv(v[i - 1], x))/(400*n^2 - 840*n + 441)); 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-aac530d688cc3150","steps":["(\u03b8+1)\u00b2"],"edges":["e-4aa333fe2966d42c"],"text":"A144442 \u2192 (\u03b8+1)\u00b2"},"headline":"A144442 \u2192 (\u03b8+1)\u00b2","chain":null}