{"id":"n-f352cbd6d9928aaa","revision":0,"first_index":0,"fingerprint":"f6-9df8a03f2559bb60","name":"theta_plus_one of Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.","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","2"],["2","10","9"],["6","52","105","60"],["24","308","1020","1260","525"],["120","2088","9912","19600","17325","5670"],["720","16056","101220","282128","393750","270270","72765"],["5040","138528","1103652","4035920","7857850","8440740","4729725","1081080"],["40320","1327392","12906648","59110480","149788100","222882660","194234040","91891800","18243225"],["362880","13999680","161884008","898623264","2833902500","5457051600","6553927380","4799995200","1964187225","344594250"],["3628800","161254080","2173602816","14272246560","54277263040","129465336000","200361241080","201443161920","127155278250","45831035250","7202019825"],["39916800","2014882560","31159410048","237487233984","1063923260400","3048436110720","5822827370360","7518709598400","6495174315630","3601009912500","1159525191825","164991726900"],["479001600","27151476480","475521950592","4143950830656","21475657803600","72250740481920","165271542195760","262639359702720","290615903978400","219958415176500","108692883198900","31623414322500","4111043861925"],["6227020800","392574712320","7703010264960","75813652302336","447965199274320","1738616747868000","4657424851839760","8841540681733760","12012319576857600","11620122893379000","7822929960846000","3486165194912400","924984868933125","110681950128750"],["87178291200","6062977267200","132081095266560","1453162974792192","9674413698362400","42709766513817312","131687834947869200","292127779501317760","473152429296827280","560897305093125000","481739428262792700","291972659756778000","118466580621138750","28887988983603750","3201870700153125"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","21","223","3137","54715","1136909","27392535","750424665","23028527987","782346067621","29145298930127","1181286063981553","51746667390073131","2436109451601402429"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-2","-1"],["2","4","2"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","2","1"],["0","-1","-2","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186040,"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-cf2c769cb0c4a355"]},{"id":186041,"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-cf2c769cb0c4a355"]},{"id":186042,"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-cf2c769cb0c4a355"]}],"region_exact":{},"incoming_edges":[{"id":"e-cf2c769cb0c4a355","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-f352cbd6d9928aaa","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":"140be2273a2c1fa11ce87446953c49d3ff1bf50ed91646af81d19793f50374ef","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-a36b2e24b7a3dbb7","role":"sequence","name":"Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.","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,        2;\n  2:      2,       10,         9;\n  3:      6,       52,       105,        60;\n  4:     24,      308,      1020,      1260,        525;\n  5:    120,     2088,      9912,     19600,      17325,       5670;\n  6:    720,    16056,    101220,    282128,     393750,     270270,      72765;\n  7:   5040,   138528,   1103652,   4035920,    7857850,    8440740,    4729725,    1081080;\n  8:  40320,  1327392,  12906648,  59110480,  149788100,  222882660,  194234040,   91891800,   18243225;\n  9: 362880, 13999680, 161884008, 898623264, 2833902500, 5457051600, 6553927380, 4799995200, 1964187225, 344594250;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(2*n^2-4*n+2)*x+(n^2-2*n+1)*x^2)*P(n-2,x) - ((n-1)*x+(2*n-2)*x^2+(n-1)*x^3)*d/dx P(n-2,x) + (2*n-1+2*n*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x + (n^2 - 2*n + 1)*x^2)*P[n - 2, x] - ((n - 1)*x + (2*n - 2)*x^2 + (n - 1)*x^3)*D[P[n - 2, x], x] + (2*n - 1 + 2*n*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x + (n^2 - 2*n + 1)*x^2)*v[i - 2] - ((n - 1)*x + (2*n - 2)*x^2 + (n - 1)*x^3)*deriv(v[i - 2], x) + (2*n - 1 + 2*n*x)*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); 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-a36b2e24b7a3dbb7","steps":["\u03b8+1"],"edges":["e-cf2c769cb0c4a355"],"text":"A112486 \u2192 \u03b8+1"},"headline":"A112486 \u2192 \u03b8+1","chain":null}