{"id":"n-ef68c61660e41606","revision":0,"first_index":0,"fingerprint":"f6-a25c0a0957a3bd5b","name":"theta_plus_one of one_plus_derivative of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= 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"],["2","2"],["4","10","3"],["5","24","21","4"],["7","56","87","40","5"],["8","82","204","180","60","6"],["10","138","495","660","375","90","7"],["11","176","792","1500","1350","594","119","8"],["13","256","1470","3696","4575","2952","994","160","9"],["14","306","2028","6232","10170","9306","4844","1392","198","10"],["16","410","3255","12280","25250","29898","20965","8720","2070","250","11"],["17","472","4155","17940","44025","65436","60641","35320","12735","2700","297","12"],["19","600","6093","30840","90300","163152","187565","139344","66960","20400","3729","360","13"],["20","674","7416","41484","137700","291060","405832","379768","239652","101100","27720","4644","416","14"],["22","826","10227","65100","248955","614376","1014398","1141976","884835","472080","171402","41076","6097","490","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","17","54","195","540","1775","4550","14125","34500","103125","243750","709375","1637500","4671875"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"],["4","10","3"],["5","24","21","4"],["7","56","87","40","5"],["8","82","204","180","60","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","9","30","45","30","9","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["-3","-18","-33","-18","-3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["3","9","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173274,"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-43f5afa97f48ec38"]},{"id":173275,"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-43f5afa97f48ec38"]},{"id":173276,"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-43f5afa97f48ec38"]}],"region_exact":{},"incoming_edges":[{"id":"e-43f5afa97f48ec38","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-ef68c61660e41606","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":"38fb24e3867b0c8732fa7bb6d0ea0c45ba86f3482181890eea5cfbc343840a81","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-582f0cc7fb344588","role":"sequence","name":"one_plus_derivative of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= 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:  2,   2;\n  2:  4,  10,    3;\n  3:  5,  24,   21,    4;\n  4:  7,  56,   87,   40,     5;\n  5:  8,  82,  204,  180,    60,    6;\n  6: 10, 138,  495,  660,   375,   90,    7;\n  7: 11, 176,  792, 1500,  1350,  594,  119,    8;\n  8: 13, 256, 1470, 3696,  4575, 2952,  994,  160,   9;\n  9: 14, 306, 2028, 6232, 10170, 9306, 4844, 1392, 198, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+9*x+30*x^2+45*x^3+30*x^4+9*x^5+x^6)*P(n-6,x) - (3+18*x+33*x^2+18*x^3+3*x^4)*P(n-4,x) + (3+9*x+3*x^2)*P(n-2,x) for n >= 6, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 4 + 10*x + 3*x^2, P(3,x) = 5 + 24*x + 21*x^2 + 4*x^3, P(4,x) = 7 + 56*x + 87*x^2 + 40*x^3 + 5*x^4, P(5,x) = 8 + 82*x + 204*x^2 + 180*x^3 + 60*x^4 + 6*x^5.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 4 + 10*x + 3*x^2;\nP[3, x_] := 5 + 24*x + 21*x^2 + 4*x^3;\nP[4, x_] := 7 + 56*x + 87*x^2 + 40*x^3 + 5*x^4;\nP[5, x_] := 8 + 82*x + 204*x^2 + 180*x^3 + 60*x^4 + 6*x^5;\nP[n_, x_] := P[n, x] = Expand[(1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*P[n - 6, x] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*P[n - 4, x] + (3 + 9*x + 3*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 4 + 10*x + 3*x^2; v[4] = 5 + 24*x + 21*x^2 + 4*x^3; v[5] = 7 + 56*x + 87*x^2 + 40*x^3 + 5*x^4; v[6] = 8 + 82*x + 204*x^2 + 180*x^3 + 60*x^4 + 6*x^5; for(n = 6, N, my(i = n + 1); v[i] = (1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*v[i - 6] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*v[i - 4] + (3 + 9*x + 3*x^2)*v[i - 2]); 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-4e0b19699f4cf0db","steps":["D+1","\u03b8+1"],"edges":["e-b1073907eb08b9b5","e-43f5afa97f48ec38"],"text":"A026374 \u2192 D+1 \u2192 \u03b8+1"},"headline":"A026374 \u2192 D+1 \u2192 \u03b8+1","chain":null}