{"id":"n-d044a3910145efa6","revision":0,"first_index":0,"fingerprint":"f6-2d7099bd5283cb79","name":"theta_plus_one_squared of basis_map_A007318 of Even row Pascal-square read by antidiagonals.","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"],["3","8"],["8","40","27"],["21","152","198","64"],["55","520","999","640","125"],["144","1680","4266","4096","1625","216"],["377","5232","16524","21184","12775","3528","343"],["987","15880","59994","96320","78250","33264","6860","512"],["2584","47288","207981","401408","410875","237168","76146","12288","729"],["6765","138760","696402","1570560","1941125","1419480","623280","158208","20655","1000"],["17711","402440","2269593","5857728","8488375","7525224","4206895","1469952","304722","33000","1331"],["46368","1156128","7238052","21041152","34986950","36478512","24813992","11089408","3183543","552200","50578","1728"],["121393","3295200","22676760","73321600","137616250","165046320","132415150","72250880","26640900","6435000","951665","74880","2197"],["317811","9329672","70000740","249192320","521132500","706890240","653847180","421893120","190535085","59345000","12285130","1572480","107653","2744"],["832040","26265160","213388047","829337600","1912386250","2895177600","3034418100","2262120960","1211105925","463385000","124149025","22351680","2506777","150920","3375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","75","435","2339","12027","59963","292067","1396467","6576235","30576971","140638611","640848195","2896451675","12997578459"],"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"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0","0"],["200","170","18","-28","0"],["0","500","160","-32","0"],["0","0","600","200","8"],["0","0","0","240","32"],["0","0","0","0","32"]]},"terms":[{"coeff":{"coeffs":[["-200","-620","-668","-284","-44","-8"],["800","1530","162","-1204","-664","-40"],["0","2700","5200","3052","428","-40"],["0","0","3000","4680","2112","248"],["0","0","0","1120","1296","320"],["0","0","0","0","128","96"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-125","-525","-885","-771","-378","-108","-16"],["0","-750","-2850","-4220","-2996","-988","-112"],["0","0","-1200","-3560","-3860","-1804","-304"],["0","0","0","-600","-1400","-1056","-256"],["0","0","0","0","-80","-144","-64"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176366,"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-1c53e23cf20ff476"]},{"id":176367,"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-1c53e23cf20ff476"]},{"id":176368,"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-1c53e23cf20ff476"]}],"region_exact":{},"incoming_edges":[{"id":"e-1c53e23cf20ff476","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-d044a3910145efa6","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":"5b0ff63e00d6f8dbfc915af0af9c9825432521d8acc37440af462781ec309682","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-74b50e0ea94d74b6","role":"sequence","name":"basis_map_A007318 of Even row Pascal-square read by antidiagonals.","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:    3,      8;\n  2:    8,     40,     27;\n  3:   21,    152,    198,      64;\n  4:   55,    520,    999,     640,     125;\n  5:  144,   1680,   4266,    4096,    1625,     216;\n  6:  377,   5232,  16524,   21184,   12775,    3528,    343;\n  7:  987,  15880,  59994,   96320,   78250,   33264,   6860,    512;\n  8: 2584,  47288, 207981,  401408,  410875,  237168,  76146,  12288,   729;\n  9: 6765, 138760, 696402, 1570560, 1941125, 1419480, 623280, 158208, 20655, 1000;\n","recurrence_sentence":"The row polynomials satisfy (200*n+(500*n^2+170*n)*x+(600*n^3+160*n^2+18*n)*x^2+(240*n^4+200*n^3-32*n^2-28*n)*x^3+(32*n^5+32*n^4+8*n^3)*x^4)*P(n,x) = (800*n-200+(2700*n^2+1530*n-620)*x+(3000*n^3+5200*n^2+162*n-668)*x^2+(1120*n^4+4680*n^3+3052*n^2-1204*n-284)*x^3+(128*n^5+1296*n^4+2112*n^3+428*n^2-664*n-44)*x^4+(96*n^5+320*n^4+248*n^3-40*n^2-40*n-8)*x^5)*P(n-1,x) - (125+(750*n+525)*x+(1200*n^2+2850*n+885)*x^2+(600*n^3+3560*n^2+4220*n+771)*x^3+(80*n^4+1400*n^3+3860*n^2+2996*n+378)*x^4+(144*n^4+1056*n^3+1804*n^2+988*n+108)*x^5+(64*n^4+256*n^3+304*n^2+112*n+16)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((800*n - 200 + (2700*n^2 + 1530*n - 620)*x + (3000*n^3 + 5200*n^2 + 162*n - 668)*x^2 + (1120*n^4 + 4680*n^3 + 3052*n^2 - 1204*n - 284)*x^3 + (128*n^5 + 1296*n^4 + 2112*n^3 + 428*n^2 - 664*n - 44)*x^4 + (96*n^5 + 320*n^4 + 248*n^3 - 40*n^2 - 40*n - 8)*x^5)*P[n - 1, x] - (125 + (750*n + 525)*x + (1200*n^2 + 2850*n + 885)*x^2 + (600*n^3 + 3560*n^2 + 4220*n + 771)*x^3 + (80*n^4 + 1400*n^3 + 3860*n^2 + 2996*n + 378)*x^4 + (144*n^4 + 1056*n^3 + 1804*n^2 + 988*n + 108)*x^5 + (64*n^4 + 256*n^3 + 304*n^2 + 112*n + 16)*x^6)*D[P[n - 1, x], x])/(200*n + (500*n^2 + 170*n)*x + (600*n^3 + 160*n^2 + 18*n)*x^2 + (240*n^4 + 200*n^3 - 32*n^2 - 28*n)*x^3 + (32*n^5 + 32*n^4 + 8*n^3)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((800*n - 200 + (2700*n^2 + 1530*n - 620)*x + (3000*n^3 + 5200*n^2 + 162*n - 668)*x^2 + (1120*n^4 + 4680*n^3 + 3052*n^2 - 1204*n - 284)*x^3 + (128*n^5 + 1296*n^4 + 2112*n^3 + 428*n^2 - 664*n - 44)*x^4 + (96*n^5 + 320*n^4 + 248*n^3 - 40*n^2 - 40*n - 8)*x^5)*v[i - 1] - (125 + (750*n + 525)*x + (1200*n^2 + 2850*n + 885)*x^2 + (600*n^3 + 3560*n^2 + 4220*n + 771)*x^3 + (80*n^4 + 1400*n^3 + 3860*n^2 + 2996*n + 378)*x^4 + (144*n^4 + 1056*n^3 + 1804*n^2 + 988*n + 108)*x^5 + (64*n^4 + 256*n^3 + 304*n^2 + 112*n + 16)*x^6)*deriv(v[i - 1], x))/(200*n + (500*n^2 + 170*n)*x + (600*n^3 + 160*n^2 + 18*n)*x^2 + (240*n^4 + 200*n^3 - 32*n^2 - 28*n)*x^3 + (32*n^5 + 32*n^4 + 8*n^3)*x^4)); 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-79ebafb266da3b41","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-1a8887aa0d003bc9","e-1c53e23cf20ff476"],"text":"A172431 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A172431 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}