{"id":"n-e14b11c6ee22ff96","revision":0,"first_index":0,"fingerprint":"f6-8adb5a6a862fea86","name":"theta_plus_one_squared of derivative of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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","8"],["2","24","27"],["6","88","162","64"],["24","400","945","640","125"],["120","2192","6075","5440","1875","216"],["720","14112","43848","47040","21875","4536","343"],["5040","104544","354564","433216","245000","69552","9604","512"],["40320","876672","3189348","4306176","2806125","979776","187278","18432","729"],["362880","8212608","31662900","46315520","33665625","13666968","3241350","445440","32805","1000"],["3628800","85029120","344346552","538208000","427116250","194843880","54116139","9292800","962280","55000","1331"],["39916800","964350720","4074785352","6736516864","5749466250","2881339560","904682394","183000576","23816430","1925000","87846","1728"],["479001600","11891543040","52152107904","90496952832","82150854500","44511152400","15431649233","3546436608","546358527","55770000","3616327","134784","2197"],["6227020800","158422072320","718111360512","1300080198144","1244712969500","720601657776","271293501479","69138674176","12152177037","1474473000","121242121","6447168","199927","2744"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","9","53","320","2134","15918","132474","1222032","12404856","137607096","1657600152","21559889520","301265579952","4502341996704"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["1","8"],["2","24","27"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-256","256","-96","16","-1"],["256","-192","48","-4","0"],["-96","48","-6","0","0"],["16","-4","0","0","0"],["-1","0","0","0","0"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-175","148","-42","4"],["148","-84","12","0"],["-42","12","0","0"],["4","0","0","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-55","36","-6"],["36","-12","0"],["-6","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-10","4"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176823,"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-677878ecda5fb949"]},{"id":176824,"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-677878ecda5fb949"]},{"id":176825,"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-677878ecda5fb949"]}],"region_exact":{},"incoming_edges":[{"id":"e-677878ecda5fb949","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-e14b11c6ee22ff96","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":"113b101fe13fad2c173de6a524f94ff5eb34c617264bde8ac62761af0b6da8cd","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-0b6da90e76b5fd69","role":"sequence","name":"derivative of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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":[{"id":"l-4c983dd84cd5f701","kind":"fingerprint_related","node_a_id":"n-e14b11c6ee22ff96","node_b_id":"n-08fcb9967597bac2","fingerprint":"f6-8adb5a6a862fea86","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8adb5a6a862fea86"},"other_id":"n-08fcb9967597bac2","other":{"id":"n-08fcb9967597bac2","name":"theta_plus_one_squared of theta_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","first_index":0,"fingerprint":"f6-8adb5a6a862fea86","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","8"],["2","24","27"]],"card_rows":[["1"],["1","8"],["2","24","27"],["6","88","162","64"]],"row_sums_preview":["1","9","53","320","2134","15918","132474","1222032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3f4ccde57d57bff1","steps":["(\u03b8+1)\u00b2"],"edges":["e-378ea302aa675ac4"],"text":"A028421 \u2192 (\u03b8+1)\u00b2"},"headline":"A028421 \u2192 (\u03b8+1)\u00b2","chain":null}}],"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:     0;\n  1:     1;\n  2:     1,      8;\n  3:     2,     24,      27;\n  4:     6,     88,     162,      64;\n  5:    24,    400,     945,     640,     125;\n  6:   120,   2192,    6075,    5440,    1875,    216;\n  7:   720,  14112,   43848,   47040,   21875,   4536,    343;\n  8:  5040, 104544,  354564,  433216,  245000,  69552,   9604,   512;\n  9: 40320, 876672, 3189348, 4306176, 2806125, 979776, 187278, 18432, 729;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256+(4*n^3-48*n^2+192*n-256)*x+(6*n^2-48*n+96)*x^2+(4*n-16)*x^3+x^4)*P(n-4,x) + (4*n^3-42*n^2+148*n-175+(12*n^2-84*n+148)*x+(12*n-42)*x^2+4*x^3)*P(n-3,x) - (6*n^2-36*n+55+(12*n-36)*x+6*x^2)*P(n-2,x) + (4*n-10+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 1 + 8*x, P(3,x) = 2 + 24*x + 27*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 1 + 8*x;\nP[3, x_] := 2 + 24*x + 27*x^2;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 + (4*n^3 - 48*n^2 + 192*n - 256)*x + (6*n^2 - 48*n + 96)*x^2 + (4*n - 16)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 42*n^2 + 148*n - 175 + (12*n^2 - 84*n + 148)*x + (12*n - 42)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 36*n + 55 + (12*n - 36)*x + 6*x^2)*P[n - 2, x] + (4*n - 10 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 1 + 8*x; v[4] = 2 + 24*x + 27*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 + (4*n^3 - 48*n^2 + 192*n - 256)*x + (6*n^2 - 48*n + 96)*x^2 + (4*n - 16)*x^3 + x^4)*v[i - 4] + (4*n^3 - 42*n^2 + 148*n - 175 + (12*n^2 - 84*n + 148)*x + (12*n - 42)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 36*n + 55 + (12*n - 36)*x + 6*x^2)*v[i - 2] + (4*n - 10 + 4*x)*v[i - 1]); 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-b9fbfb57d9269e2f","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-55ee0b1ee1acb060","e-677878ecda5fb949"],"text":"A132393 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A132393 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}